Answers

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pyri

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he M

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Inc.

Chapter 10 A1 Glencoe Algebra 1

Chapter Resources

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

3

Gle

ncoe A

lgeb

ra 1

An

tici

pati

on

Gu

ide

Rad

ical

Exp

ressio

ns a

nd

Tri

an

gle

s

Befo

re y

ou

beg

in C

ha

pte

r 1

0

•

R

ead

each

sta

tem

ent.

•

D

ecid

e w

het

her

you

Agre

e (A

) or

Dis

agre

e (D

) w

ith

th

e st

ate

men

t.

•

W

rite

A o

r D

in

th

e fi

rst

colu

mn

OR

if

you

are

not

su

re w

het

her

you

agre

e or

d

isagre

e, w

rite

NS

(N

ot S

ure

).

ST

EP

1A

, D

, o

r N

SS

tate

men

tS

TE

P 2

A o

r D

1.

An

exp

ress

ion

th

at

con

tain

s a s

qu

are

roo

t is

call

ed a

ra

dic

al

exp

ress

ion

.A

2.

It i

s alw

ays

tru

e th

at

√ �

xy w

ill

equ

al

√ � x

· √

� y .A

3.

1

−

√ �

3

is

in s

imp

lest

for

m b

ecau

se √

�

3 i

s n

ot a

wh

ole

nu

mber

.D

4.

Th

e su

m o

f 3

√ �

3

an

d 2

√ �

3

wil

l eq

ual

5 √

�

3 .

A

5.

Bef

ore

mu

ltip

lyin

g t

wo

rad

ical

exp

ress

ion

s w

ith

dif

fere

nt

rad

ican

ds

the

squ

are

roo

ts m

ust

be

evalu

ate

d.

D

6.

Wh

en s

olvin

g r

ad

ical

equ

ati

ons

by s

qu

ari

ng e

ach

sid

e of

th

e eq

uati

on,

it i

s p

ossi

ble

to

obta

in s

olu

tion

s th

at

are

not

so

luti

ons

to t

he

orig

inal

equ

ati

on.

A

7.

Th

e lo

nges

t si

de

of a

ny t

rian

gle

is

call

ed t

he

hyp

oten

use

.D

8.

Bec

au

se 5

2 =

42 +

32,

a t

rian

gle

wh

ose

sid

es h

ave

len

gth

s 3,

4,

an

d 5

wil

l be

a r

igh

t tr

ian

gle

.A

9.

On

a c

oord

inate

pla

ne,

th

e d

ista

nce

bet

wee

n a

ny t

wo

poi

nts

ca

n b

e fo

un

d u

sin

g t

he

Pyth

agor

ean

Th

eore

m.

A

10.

Th

e D

ista

nce

For

mu

la c

an

not

be

use

d t

o fi

nd

th

e d

ista

nce

bet

wee

n t

wo

poi

nts

on

th

e sa

me

ver

tica

l li

ne.

D

11.

Tw

o tr

ian

gle

s are

sim

ilar

only

if

thei

r co

rres

pon

din

g a

ngle

s are

con

gru

ent

an

d t

he

mea

sure

s of

thei

r co

rres

pon

din

g s

ides

are

in

pro

por

tion

.A

12.

All

rig

ht

tria

ngle

s are

sim

ilar.

D

A

fter y

ou

com

ple

te C

ha

pte

r 1

0

•

R

erea

d e

ach

sta

tem

ent

an

d c

omp

lete

th

e la

st c

olu

mn

by e

nte

rin

g a

n A

or

a D

.

•

D

id a

ny o

f you

r op

inio

ns

abou

t th

e st

ate

men

ts c

han

ge

from

th

e fi

rst

colu

mn

?

•

F

or t

hos

e st

ate

men

ts t

hat

you

mark

wit

h a

D,

use

a p

iece

of

pap

er t

o w

rite

an

ex

am

ple

of

wh

y y

ou d

isagre

e.

10 Ste

p 1

Ste

p 2

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

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D

Lesson 10-1

Ch

ap

ter

10

5

Gle

ncoe A

lgeb

ra 1

Dilati

on

s o

f R

ad

ical

Fu

nct

ion

s A

sq

ua

re r

oo

t fu

ncti

on

con

tain

s th

e sq

uare

roo

t of

a v

ari

able

. S

qu

are

roo

t fu

nct

ion

s are

a t

yp

e of

ra

dic

al

fun

cti

on

.

In o

rder

for

a s

qu

are

roo

t to

be

a r

eal

nu

mber

, th

e ra

dic

an

d,

or t

he

exp

ress

ion

un

der

th

e ra

dic

al

sign

, ca

nn

ot b

e n

egati

ve.

Valu

es t

hat

mak

e th

e ra

dic

an

t n

egati

ve

are

not

in

clu

ded

in

th

e d

omain

.

Sq

uare

Ro

ot

Fu

ncti

on

Pare

nt

function:

f(x)

= √

� x

Type o

f gra

ph:

curv

e

Dom

ain

: {x

|x ≥

0}

Range:

{y|

y ≥

0}

G

ra

ph

y =

3 √

�x .

Sta

te t

he d

om

ain

an

d r

an

ge.

Ste

p 1

Mak

e a t

able

. C

hoo

se n

onn

egati

ve

S

tep

2 P

lot

poi

nts

an

d d

raw

a

valu

es f

or x

.

smoo

th c

urv

e.

xy

00

0.5

≈ 2

.12

13

2≈

4.2

4

46

6≈

7.3

5

y

x

y=

3 x

Th

e d

omain

is

{x|

x ≥

0}

an

d t

he

ran

ge

is {

y|

y ≥

0}.

Exerc

ises

Gra

ph

ea

ch

fu

ncti

on

, a

nd

co

mp

are t

o t

he p

aren

t g

ra

ph

. S

tate

th

e d

om

ain

an

d r

an

ge.

1. y =

3

−

2 √

� x 2. y =

4 √

� x

3. y =

5

−

2 √

� x

y

x

y

x

y

x

D

ilati

on

of

y =

√ " x ;

D

ilati

on

of

y =

√ " x ;

D

ilati

on

of

y =

√ " x ;

D

= {

x |

x ≥

0};

D

= {

x |

x ≥

0};

D

= {

x |

x ≥

0};

R

= {

y |

y ≥

0}

R

= {

y |

y ≥

0}

R =

{y |

y ≥

0}

10-1

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Sq

uare

Ro

ot

Fu

ncti

on

s

Exam

ple

y

x

y=

x

Answers (Anticipation Guide and Lesson 10-1)

Co

pyrig

ht ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f Th

e M

cG

raw

-Hill C

om

pan

ies, In

c.

Chapter 10 A2 Glencoe Algebra 1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

6

Gle

ncoe A

lgeb

ra 1

Refle

ctio

ns a

nd

Tran

sla

tio

ns o

f R

ad

ical

Fu

nctio

ns

Rad

ical

fun

ctio

ns,

lik

e

qu

ad

rati

c fu

nct

ion

s, c

an

be t

ran

slate

d h

ori

zon

tall

y a

nd

vert

icall

y,

as

well

as

refl

ect

ed

acr

oss

the x

-axis

. T

o d

raw

th

e g

rap

h o

f y =

a √ """

x +

h ,

foll

ow

th

ese

ste

ps.

Gra

ph

s o

f S

qu

are

Ro

ot

Fu

ncti

on

s

Ste

p 1

D

raw

th

e g

rap

h o

f y =

+c √ x .

Th

e g

rap

h s

tart

s at

the

ori

gin

an

d p

ass

es

thro

ugh

th

e p

oin

t at

(1,

a).

If

a > 0

, th

e

gra

ph

is

in t

he 1

st q

uad

ran

t. I

f a

< 0

, th

e g

rap

h i

s re

flect

ed

acr

oss

th

e x

-axis

an

d i

s in

th

e 4

th q

uad

ran

t.

Ste

p 2

T

ran

slate

th

e g

rap

h

c u

nit

s u

p i

f c

is p

osi

tive a

nd

dow

n

if c

is

negati

ve.

Ste

p 3

T

ran

slate

th

e g

rap

h

h

un

its

left

if

h i

s p

osi

tive a

nd

rig

ht

if h

is

negati

ve.

G

ra

ph

y =

- √ """

x + 1

an

d c

om

pa

re t

o t

he p

aren

t g

ra

ph

. S

tate

th

e

do

ma

in a

nd

ra

ng

e.

Ste

p 1

M

ak

e a

table

of

valu

es.

x-

1

01

3

8

y 0

-1

-1.4

1-

2-

3

Ste

p 2

T

his

is

a h

ori

zon

tal

tran

slati

on

1 u

nit

to t

he l

eft

of

the p

are

nt

fun

ctio

n a

nd

refl

ect

ed

acr

oss

th

e x

-axis

. T

he d

om

ain

is {x

| x ≥

0}

an

d t

he r

an

ge i

s {y

| y ≤

0}.

Exerc

ises

Gra

ph

ea

ch

fu

ncti

on

, a

nd

co

mp

are t

o t

he p

aren

t g

ra

ph

. S

tate

th

e d

om

ain

an

d r

an

ge.

1. y =

√ " x +

3

2.

y = √ """

x -

1

3.

y =

- √ """

x -

1

y

x

y

x

y

x

10-1

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n (c

on

tin

ued

)

Sq

uare

Ro

ot

Fu

ncti

on

s

Exam

ple

y

x

y=

x

y= -

x+

1

tran

sla

tio

n o

f y =

√ "

x

up

3 u

nit

s;

D =

{x |

x ≥

0};

R =

{y |

y ≥

3}

tran

sla

tio

n o

f y =

√ "

x

rig

ht

1 u

nit

;D

= {

x |

x ≥

1};

R =

{y |

y ≥

0}

tran

sla

tio

n o

f y =

√ "

x

rig

ht

1 u

nit

an

d r

efl

ecte

d

acro

ss t

he x

-axis

;D

= {

x |

x ≥

1}

R = {

y |

y ≤

0}

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Lesson 10-1

Ch

ap

ter

10

7

Gle

ncoe A

lgeb

ra 1

Gra

ph

ea

ch

fu

ncti

on

, a

nd

co

mp

are t

o t

he p

aren

t g

ra

ph

. S

tate

th

e d

om

ain

an

d r

an

ge.

1. y =

2 √ " x

2. y =

1 −

2 √ " x

3. y =

5 √ " x

dilati

on

of

y =

√ x ;

dil

ati

on

of

y =

√ x ;

dil

ati

on

of

y =

√ x ;

D =

{x |

x ≥

0},

R =

{ y

| y

≥ 0

}

D

= {

x |

x ≥

0},

R =

{ y

| y

≥ 0

}

D =

{x |

x ≥

0},

R =

{ y

| y

≥ 0

}

4. y =

√ " x +

1

5. y =

√ " x - 4

6. y =

√ """

x - 1

tran

sla

tio

n o

f

tran

sla

tio

n o

f t

ran

sla

tio

n o

f

y =

√ " x

up

1 u

nit

; y

= √ x

do

wn

4 u

nit

s;

y =

√ x

rig

ht

1 u

nit

;

D =

{x |

x ≥

0},

R =

{ y

| y

≥ 1

}

D =

{x |

x ≥

0},

R =

{ y

| y

≥ -

4}

D =

{x |

x ≥

1},

R =

{ y

| y

≥ 0

}

7. y =

- √ """

x -

3

8. y =

√ """

x -

2 + 3

9. y =

- 1

−

2 √ """

x - 4

+ 1

tran

sla

tio

n o

f y =

√ " x ;

tr

an

sla

tio

n o

f y =

√ x

dia

lati

on

of

y =

√ " x

refl

ecte

d

rig

ht

3 u

nit

s r

efl

ecte

d a

cro

ss

rig

ht

2 u

nit

s a

nd

up

a

cro

ss t

he x

-axis

the x

-axis

; D

= {

x |

x ≥

3},

3 u

nit

s;

D =

{x |

x ≥

2},

tran

sla

ted

rig

ht

4 u

nit

s u

p

R

= {

y |

y ≤

0}

R =

{y |

y ≥

3}

1 u

nit

s;

D =

{x |

x ≥

4},

R

= {

y |

y ≤

1}

10-1

Sk

ills

Pra

ctic

e

Sq

uare

Ro

ot

Fu

ncti

on

s

y

x

y

x

y

x

12 8 4

−2

24

y

x

y

x

y

x

y

x

y

x

y

x

Answers (Lesson 10-1)

Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he M

cG

raw

-Hill C

om

pan

ies,

Inc.

Chapter 10 A3 Glencoe Algebra 1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

8

Gle

ncoe A

lgeb

ra 1

Gra

ph

ea

ch

fu

ncti

on

, a

nd

co

mp

are t

o t

he p

aren

t g

ra

ph

. S

tate

th

e d

om

ain

an

d r

an

ge.

1. y =

4 −

3 √ # x

2. y =

√ # x +

2

3. y =

√ ###

x - 3

y

x

y

x

y

x

4. y =

- √ # x +

1

5. y =

2 √

x - 1

+ 1

6. y =

- √

x -

2 +

2

y

x

y

x

y

x

7. O

HM

’S L

AW

In

ele

ctri

cal

en

gin

eeri

ng,

the r

esi

stan

ce o

f a c

ircu

it

can

be f

ou

nd

by t

he e

qu

ati

on

I =

√

P −

R , w

here

I i

s th

e c

urr

en

t in

am

pere

s, P

is

the p

ow

er

in w

att

s, a

nd

R i

s th

e r

esi

stan

ce o

f

the c

ircu

it i

n o

hm

s. G

rap

h t

his

fu

nct

ion

for

a c

ircu

it w

ith

a

resi

stan

ce o

f 4 o

hm

s.

10-1

Practi

ce

Sq

uare

Ro

ot

Fu

ncti

on

s

Current (amperes)

23 1 020

45

Pow

er (

wat

ts)

40

60

80

100

dilati

on

of

y =

√ "

x ;

D =

{x |

x ≥

0},

R =

{y |

y ≥

0}

tran

sla

tio

n o

f y =

√ "

x

up

2 u

nit

;D

= {

x |

x ≥

0},

R

= {

y |

y ≥

2}

tran

sla

tio

n o

f y =

√ "

x

left

3 u

nit

s;

D = {

x |

x ≥

-3},

R

= {

y |

y ≥

0}

tran

sla

tio

n o

f y =

√ "

x

up

1 u

nit

refl

ecte

d

in t

he x

-axis

; D

= {

x |

x ≥

0},

R =

{y |

y ≤

1}

dilati

on

of

y = √ "

x

tran

sla

ted

up

1 u

nit

an

d r

igh

t 1 u

nit

;D

= {

x |

x ≥

1},

R =

{y |

y ≥

1}

tran

sla

tio

n o

f y =

√ "

x ;

u

p 2

un

its a

nd

rig

ht

2 u

nit

s,

refl

ecte

d

in t

he x

-axis

; D

= {

x |

x ≥

2},

R =

{y |

y ≤

2}

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Lesson 10-1

Ch

ap

ter

10

9

Gle

ncoe A

lgeb

ra 1

10-1

Wo

rd

Pro

ble

m P

racti

ce

Sq

uare

Ro

ot

Fu

ncti

on

s

1. PEN

DU

LU

M M

OTIO

N T

he p

eri

od

T o

f

a p

en

du

lum

in

seco

nd

s, w

hic

h i

s th

e t

ime

for

the p

en

du

lum

to r

etu

rn t

o t

he p

oin

t

of

rele

ase

, is

giv

en

by t

he e

qu

ati

on

T=

1.1

1 √# L

. T

he l

en

gth

of

the

pen

du

lum

in

feet

is g

iven

by L

. G

rap

h

this

fu

nct

ion

.

2. EM

PIR

E S

TA

TE B

UIL

DIN

G T

he r

oof

of

the E

mp

ire S

tate

Bu

ild

ing i

s 1250 f

eet

above t

he g

rou

nd

. T

he v

elo

city

of

an

obje

ct d

rop

ped

fro

m a

heig

ht

of

h m

ete

rs

is g

iven

by t

he f

un

ctio

n V

= √ ##

2gh

,

wh

ere

g i

s th

e g

ravit

ati

on

al

con

stan

t,

32.2

feet

per

seco

nd

squ

are

d.

If a

n o

bje

ct

is d

rop

ped

fro

m t

he r

oof

of

the b

uil

din

g,

how

fast

is

it t

raveli

ng w

hen

it

hit

s th

e

stre

et

belo

w?

ap

pro

xim

ate

ly 2

84 f

t/s

3. ER

RO

R A

NA

LY

SIS

G

regory

is

dra

win

g

the g

rap

h o

f y =

-5 √ ###

x +

1 .

He d

esc

ribes

the r

an

ge a

nd

dom

ain

as

{x %

x ≥ -

1},

{y %

y ≥ 0

}. E

xp

lain

an

d c

orr

ect

th

e

mis

tak

e t

hat

Gre

gory

mad

e.

T

he d

om

ain

is a

ctu

ally {

y %

y ≤

0}

becau

se t

he g

rap

h h

as b

een

re

flecte

d a

cro

ss t

he x

-axis

.

4. C

APA

CIT

OR

S A

cap

aci

tor

is a

set

of

pla

tes

that

can

sto

re e

nerg

y i

n a

n

ele

ctri

c fi

eld

. T

he v

olt

age V

requ

ired

to

store

E j

ou

les

of

en

erg

y i

n a

cap

aci

tor

wit

h a

cap

aci

tan

ce o

f C

fara

ds

is g

iven

by V=√

2

E− C

.

a

. R

ew

rite

an

d s

imp

lify

th

e e

qu

ati

on

for

the c

ase

of

a 0

.0002 f

ara

d c

ap

aci

tor.

V =

100 √

E

b

. G

raph

th

e e

qu

ati

on

you

fou

nd i

n p

art

a.

Voltage (volts)

100

150

50 0

2

200

250

300

350

Ener

gy (

joule

s)

46

810

c.

How

wou

ld t

he g

rap

h d

iffe

r if

you

wis

hed

to s

tore

E +

1 j

ou

les

of

en

erg

y

in t

he c

ap

aci

tor

inst

ead

?

tran

sla

tio

n o

f V=

100 √" E

o

ne u

nit

to

th

e l

eft

d

. H

ow

wou

ld t

he g

rap

h d

iffe

r if

you

ap

pli

ed

a v

olt

age o

f V

+ 1

volt

s

inst

ead

?

tran

sla

tio

n o

fV=

100 √" E

o

ne u

nit

do

wn

Period (sec)

23 1 04

45

Pen

dulu

m L

engt

h (

ft)

812

16

20

Answers (Lesson 10-1)

Co

pyrig

ht ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f Th

e M

cG

raw

-Hill C

om

pan

ies, In

c.

Chapter 10 A4 Glencoe Algebra 1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

10

Gle

ncoe A

lgeb

ra 1

A c

ub

ic r

oo

t fu

ncti

on

con

tain

s th

e c

ubic

root

of

a v

ari

able

. T

he c

ub

ic r

oo

t of

a n

um

ber

x a

re t

he n

um

bers

y t

hat

sati

sfy t

he e

qu

ati

on

y ·

y ·

y =

x (

or,

alt

ern

ati

vely

, y =

3 √ " x

).U

nli

ke s

qu

are

root

fun

ctio

ns,

cu

bic

root

fun

ctio

ns

retu

rn r

eal

nu

mbers

wh

en

th

e r

ad

ican

d

is n

egati

ve.

G

ra

ph

y =

3 √ "

x .

Ste

p 1

Mak

e a

table

. S

tep

2 P

lot

poin

ts a

nd

dra

w a

sm

ooth

cu

rve.

xy

−5

−1.7

1

−3

−1.4

4

−1

−1

00

11

31.4

4

51.7

1

y

x

Exerc

ises

Gra

ph

ea

ch

fu

ncti

on

, a

nd

co

mp

are t

o t

he p

aren

t g

ra

ph

.

1. y =

2 3

√ " x

2. y =

3 √ " x +

1

3. y =

3 √ """

x +

1

y

x

y

x

y

x

d

ilati

on

of

y =

3 √ " x

tra

nsla

tio

n o

f t

ran

sla

tio

n o

f

y

=

3 √ " x

up

1 u

nit

y

=

3 √ " x

left

1 u

nit

4. y =

3 √ """

x -

1 +

2

5. y = 3

3 √ """

x -

2

6. y = - 3

√ " x +

3

y

x

y

x

y

x

tr

an

sla

tio

n o

f y =

3 √ " x

dilati

on

of

refl

ecti

on

of

y =

3 √ " x

u

p 2

un

its a

nd

y

= 3 √ " x

tran

sla

ted

a

cro

ss t

he x

-axis

r

igh

t 1 u

nit

r

igh

t 2 u

nit

s

tra

nsla

ted

up

3 u

nit

s

10-1

En

rich

men

t

Cu

bic

Ro

ot

Fu

ncti

on

s

Exam

ple

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Lesson 10-2

Ch

ap

ter

10

11

Gle

ncoe A

lgeb

ra 1

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Sim

plify

ing

Rad

ical

Exp

ressio

ns

Pro

du

ct P

rop

ert

y o

f Sq

uare

Ro

ots

T

he P

ro

du

ct

Pro

perty

of

Sq

ua

re R

oo

ts a

nd

p

rim

e f

act

ori

zati

on

can

be u

sed

to s

imp

lify

exp

ress

ion

s in

volv

ing i

rrati

on

al

squ

are

roots

. W

hen

you

sim

pli

fy r

ad

ical

exp

ress

ion

s w

ith

vari

able

s, u

se a

bso

lute

valu

e t

o e

nsu

re

non

negati

ve r

esu

lts.

S

imp

lify

√ ""

180 .

√ ""

180 =

√ """"""

2 #

2 #

3 #

3 #

5

Prim

e f

acto

rization o

f 180

= √ "

22 #

√ "

32 #

√ "

5

Pro

duct

Pro

pert

y o

f S

quare

Roots

= 2

# 3

# √ "

5

Sim

plif

y.

= 6

√ "

5

Sim

plif

y.

S

imp

lify

√ """"""

120

a2 ·

b5 ·

c4 .

√ """"""

120

a2 #

b5 #

c4

=

√ """"""""

23 #

3 #

5 #

a2 #

b5 #

c4

=

√ "

22 #

√ "

2 #

√ "

3 #

√ "

5 #

√ "

a2 #

√ """

b4 #

b #

√ "

c4

=

2 #

√ "

2 #

√ "

3 #

√ "

5 #

a #

b2 #

√ "

b #

c2

=

2 a b

2c2

√ ""

30b

Exerc

ises

Sim

pli

fy e

ach

ex

pressio

n.

1. √ ""

28

2. √ ""

68

3. √ ""

60

4. √ ""

75

2 √ "

7

2 √ ""

17

2 √ ""

15

5 √ "

3

5. √ ""

162

6. √ "

3 ·

√ "

6

7. √ "

2 ·

√ "

5

8. √ "

5 ·

√ ""

10

9 √ "

2

3 √ "

2

√ ""

10

5 √ "

2

9. √ ""

4a

2

10. √ ""

9x

4

11. √ """

300

a4

12. √ """

128

c6

2 a

3x

2

10

a2 √ "

3

8 c

3 √ "

2

13. 4 √ ""

10 #

3 √ "

6

14. √ ""

3x

2 #

3 √ ""

3x

4

15. √ """

20

a2b

4

16. √ """

100

x3y

24 √ ""

15

9 x

3

2 a

b2 √ "

5

10

x √ "

xy

17. √ """

24

a4b

2

18. √ """

81

x4y

2

19. √ """"

150

a2b

2c

2a

2 b

√ "

6

9x

2 y

5 a

b √ ""

6c

20. √ """"

72

a6b

3c2

21. √ """"

45

x2y

5z8

22. √ """"

98

x4y

6z2

6 a

3b

c √ ""

2b

3

x y

2z

4 √ ""

5y

7x

2 y

3z √ "

2

10-2

Pro

du

ct

Pro

pert

y o

f S

qu

are

Ro

ots

For

any n

um

bers

a a

nd b

, w

here

a ≥

0 a

nd b

≥ 0

, √ ""

ab =

√ " a # √ "

b .

Exam

ple

1

Exam

ple

2

Answers (Lesson 10-1 and Lesson 10-2)

Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he M

cG

raw

-Hill C

om

pan

ies,

Inc.

Chapter 10 A5 Glencoe Algebra 1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

12

Gle

ncoe A

lgeb

ra 1

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n (c

on

tin

ued

)

Sim

plify

ing

Rad

ical

Exp

ress

ion

s

Qu

oti

en

t Pro

pert

y o

f Sq

uare

Ro

ots

A

fra

ctio

n c

on

tain

ing r

ad

icals

is

in s

imp

lest

fo

rm i

f n

o r

ad

icals

are

left

in

th

e d

en

om

inato

r. T

he Q

uo

tien

t P

ro

perty

of

Sq

ua

re R

oo

ts

an

d r

ati

on

ali

zin

g t

he d

en

om

ina

tor c

an

be u

sed

to s

imp

lify

rad

ical

exp

ress

ion

s th

at

involv

e d

ivis

ion

. W

hen

you

rati

on

ali

ze t

he d

en

om

inato

r, y

ou

mu

ltip

ly t

he n

um

era

tor

an

d

den

om

inato

r by a

rad

ical

exp

ress

ion

th

at

giv

es

a r

ati

on

al

nu

mber

in t

he d

en

om

inato

r.

S

imp

lify

√ �

�

56

−

45 .

√ ��

56

−

45 =

√ �

��

4 "

14

−

9 "

5

=

2 "

√ �

�

14

−

3 "

√ �

�

15

Sim

plif

y t

he n

um

era

tor

and d

enom

inato

r.

=

2 √

��

14

−

3 √

�

5

" √

�

5

−

√ �

5

Multip

ly b

y √

�

5

−

√ �

5 t

o r

ationaliz

e t

he d

enom

inato

r.

=

2 √

��

70

−

15

P

roduct

Pro

pert

y o

f S

quare

Roots

Exerc

ises

Sim

pli

fy e

ach

ex

pressio

n.

1. √

�

9

−

√ �

18

√

�

2

−

2

2. √

�

8

−

√ �

24

√

� 3

−

3

3. √

��

100

−

√ �

�

121

1

0

−

11

4. √

�

75

−

√ �

3

5

5. 8

√ �

2

−

2 √

�

8

2

6. √ �

2

−

5 "

√ �

6

−

5

2

√ � 3

−

5

7. √ �

3

−

4 "

√ �

5

−

2

√

��

30

−

4

8. √ �

5

−

7 "

√ �

2

−

5

√

��

14

−

7

9. √ �

�

3a

2

−

10b

6

|a

| √ ��

30

−

10| b

3 |

10. √ �

x

6

−

y4

x3

−

y2

11. √ �

�

100a

4

−

144b

8

5

a2

−

6b

4

12. √ �

��

75b

3c6

−

a

2

5

bc

3 √

��

3b

−

a

13.

√ �

4

−

3 -

√ �

5

3

+ √

�

5

−

2

14.

√ �

8

−

2 +

√ �

3

4 √

�

2 -

2 √

� 6

15.

√ �

5

−

5 +

√ �

5

5 -

2 √

�

5

16.

√ �

8

−

2 √

�

7 +

4 √

�

10

4

√ �

5 -

√ ��

14

−

33

10-2

Qu

oti

en

t P

rop

ert

y o

f S

qu

are

Ro

ots

For

any n

um

bers

a a

nd b

, w

here

a ≥

0 a

nd b

> 0

, √ �

a

−

b =

√

�

a

−

√ �

b

.

Exam

ple

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Lesson 10-2

Ch

ap

ter

10

13

Gle

ncoe A

lgeb

ra 1

Sk

ills

Pra

ctic

e

Sim

plify

ing

Rad

ical

Exp

ressio

ns

Sim

pli

fy e

ach

ex

pressio

n.

1.

√ �

�

28

2 √

�

7

2.

√ �

�

40

2 √

��

10

3.

√ �

�

72

6 √

�

2

4.

√ �

�

99

3 √

��

11

5.

√ �

2 ·

√ �

�

10

2 √

�

5

6.

√ �

5 ·

√ �

�

60

10 √

�

3

7.

3 √

�

5 ·

√ �

5

15

8.

√ �

6 ·

4 √

��

24

48

9.

2 √

�

3 ·

3 √

��

15

18 √

�

5

10.

√ �

�

16

b4

4

b2

11.

√ �

��

81

a2d

4

9| a

|d2

12.

√ �

��

40

x4y

6

2x

2 y

3 √

��

10

13.

√ �

��

75

m5P

2

5

m2| P

| √ �

�

3m

14.

√ �

5

−

3

√

��

15

−

3

15.

√ �

1

−

6

√

�

6

−

6

16.

√ �

6

−

7 ·

√ �

1

−

3

√

��

14

−

7

17.

√ ��

q

−

12

√

��

3q

−

6

18.

√ ��

4h

−

5

2

√ �

�

5h

−

5

19.

√ ��

12

−

b2

2

√ �

3

−

| b |

20.

√ ��

45

−

4m

4

3

√ �

5

−

2m

2

21.

2

−

4 +

√ �

5

8

- 2

√ �

5

−

11

22.

3

−

2 -

√ �

3

6 +

3 √

�

3

23.

5

−

7 +

√ �

7

3

5 -

5 √

�

7

−

42

24.

4

−

3 -

√ �

2 1

2 +

4 √

�

2

−

7

10-2

Answers (Lesson 10-2)

Co

pyrig

ht ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f Th

e M

cG

raw

-Hill C

om

pan

ies, In

c.

Chapter 10 A6 Glencoe Algebra 1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

14

Gle

ncoe A

lgeb

ra 1

Practi

ce

Sim

plify

ing

Rad

ical

Exp

ressio

ns

Sim

pli

fy.

1.

√ �

24

2 √

� 6

2.

√ �

60

2 √

��

15

3. √

��

108 6

√ � 3

4. √

�

8 !

√ �

6 4

√ � 3

5. √

�

7 !

√ �

14 7 √

�

2

6. 3

√ �

12 !

5 √

�

6 9

0 √

�

2

7. 4 √

�

3 !

3 √

�

18 3

6 √

� 6

8. √

��

�

27tu

3

3 u

√ �

�

3tu

9. √

��

50p

5 5

p2 √

��

2p

10. √

��

��

108x

6y

4z5

6

x3 y

2z

2 √

��

3z

11. √

��

��

56m

2n

4p

5 2

m n

2p

2 √

��

14

p

12. √

�

8

−

√ �

6

2

√ �

3

−

3

13. √ �

2

−

10

√ �

5

−

5

14. √ �

5

−

32

√

��

10

−

8

15. √ �

3

−

4 !

√ �

4

−

5

√

��

15

−

5

16. √ �

1

−

7 !

√ �

7

−

11

√

��

11

−

11

17. √

�

3k

−

√ �

8

√

��

6k

−

4

18. √ �

1

8

−

x3

3

√ ��

2x

−

x2

19. √ �

�

4y

−

3y

2

20. √ �

�

9ab

−

4ab

4

3

√ �

b

−

2

b2

21.

3

−

5 -

√ �

2

15 +

3 √

�

2

−

23

22.

8

−

3 +

√ �

3

1

2 -

4 √

�

3

−

3

23.

5

−

√ �

7 +

√ �

3

5

√ �

7 -

5 √

�

3

−

4

24.

3 √

�

7

−

-1 -

√ �

27

3

√ �

7 -

9 √

��

21

−

26

25. S

KY

DIV

ING

W

hen

a s

kyd

iver

jum

ps

from

an

air

pla

ne,

the t

ime t

it

tak

es

to f

ree f

all

a

giv

en

dis

tan

ce c

an

be e

stim

ate

d b

y t

he f

orm

ula

t =

√ �

�

2s

−

9.8

, w

here

t i

s in

seco

nd

s an

d s

is

in

mete

rs.

If J

uli

e j

um

ps

from

an

air

pla

ne,

how

lon

g w

ill

it t

ak

e h

er

to f

ree f

all

750

mete

rs?

26. M

ET

EO

RO

LO

GY

T

o e

stim

ate

how

lon

g a

th

un

ders

torm

wil

l la

st,

mete

oro

logis

ts c

an

use

the f

orm

ula

t =

√ �

�

d3

−

216 ,

wh

ere

t i

s th

e t

ime i

n h

ou

rs a

nd

d i

s th

e d

iam

ete

r of

the s

torm

in

m

iles.

a

. A

th

un

ders

torm

is

8 m

iles

in d

iam

ete

r. E

stim

ate

how

lon

g t

he s

torm

wil

l la

st.

Giv

e y

ou

r an

swer

in s

imp

lifi

ed

form

an

d a

s a d

eci

mal.

b

. W

ill

a t

hu

nd

ers

torm

tw

ice t

his

dia

mete

r la

st t

wic

e a

s lo

ng?

Exp

lain

.

N

o;

it w

ill

last

ab

ou

t 4.4

h,

or

nearl

y 3

tim

es a

s l

on

g.

10-2

ab

ou

t 12.4

s

8

√ �

3

−

9

h ≈

1.5

h

2 √

��

3y

−

3 y

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Lesson 10-2

Ch

ap

ter

10

15

Gle

ncoe A

lgeb

ra 1

Wo

rd

Pro

ble

m P

racti

ce

Sim

plify

ing

Rad

ical

Exp

ressio

ns

1. S

PO

RT

S Jasm

ine c

alc

ula

ted

th

e h

eig

ht

of

her

team

’s s

occ

er

goal

to b

e

15− √

3 f

eet.

S

imp

lify

th

e e

xp

ress

ion

.

5 √

�

3

2. N

AT

UR

E In

2004,

an

eart

hqu

ak

e b

elo

w

the o

cean

flo

or

init

iate

d a

devast

ati

ng

tsu

nam

i in

th

e I

nd

ian

Oce

an

. S

cien

tist

s ca

n a

pp

roxim

ate

th

e v

elo

city

(in

feet

per

seco

nd

) of

a t

sun

am

i in

wate

r of

dep

th d

(in

feet)

wit

h t

he f

orm

ula

V =

√ ##

16d

. D

ete

rmin

e t

he v

elo

city

of

a t

sun

am

i in

300 f

eet

of

wate

r. W

rite

you

r an

swer

in

sim

pli

fied

rad

ical

form

.

40 √

�

3 f

t/s

3. A

UT

OM

OB

ILE

S T

he f

oll

ow

ing f

orm

ula

ca

n b

e u

sed

to f

ind

th

e “

zero

to s

ixty

” ti

me f

or

a c

ar,

or

the t

ime i

t ta

kes

for

a

car

to a

ccele

rate

fro

m a

sto

p t

o s

ixty

m

iles

per

hou

r. V =

√ ##

2P

T −

M

V

is

the v

elo

city

(in

mete

rs p

er

seco

nd

).P

is

the c

ar’

s avera

ge p

ow

er

(in

watt

s).

M i

s th

e m

ass

of

the c

ar

(in

kil

ogra

ms)

.T

is

the t

ime (

in s

eco

nd

s).

F

ind

th

e t

ime i

t ta

kes

for

a 9

00-k

ilogra

m

car

wit

h a

n a

vera

ge 6

0,0

00 w

att

s of

pow

er

to a

ccele

rate

fro

m s

top

to 2

6.8

2

mete

rs p

er

seco

nd

(60 m

iles

per

hou

r).

Rou

nd

you

r an

swer

to t

he n

eare

st t

en

th.

ab

ou

t 5.4

s

4. P

HY

SIC

AL S

CIE

NC

E W

hen

a s

ubst

an

ce

such

as

wate

r vap

or

is i

n i

ts g

ase

ou

s st

ate

, th

e v

olu

me a

nd

th

e v

elo

city

of

its

mole

cule

s in

crease

as

tem

pera

ture

in

crease

s. T

he a

vera

ge v

elo

city

V o

f a

mole

cule

wit

h m

ass

m a

t te

mp

era

ture

T

is g

iven

by t

he f

orm

ula

V= √##

3kT−

m.

Solv

e t

he e

qu

ati

on

for

k.

k

=m

V 2

− 3T

5. G

EO

ME

TR

Y

Su

pp

ose

Em

ery

vil

le

Hosp

ital

wan

ts t

o b

uil

d a

new

h

eli

pad

on

wh

ich

med

ic r

esc

ue

heli

cop

ters

can

lan

d.

Th

e

heli

pad

wil

l be c

ircu

lar

an

d

mad

e o

f fi

re r

esi

stan

t ru

bber.

a

. If

th

e a

rea o

f th

e h

eli

pad

is

A,

wri

te a

n

equ

ati

on

for

the r

ad

ius

r.

r

= √

�

A

−

π

b

. W

rite

an

exp

ress

ion

in

sim

pli

fied

ra

dic

al

form

for

the r

ad

ius

of

a h

eli

pad

w

ith

an

are

a o

f 288 s

qu

are

mete

rs.

r

= 1

2 √

��

2π

−

π

c.

Usi

ng y

ou

r ca

lcu

lato

r, f

ind

a d

eci

mal

ap

pro

xim

ati

on

for

the r

ad

ius.

Rou

nd

you

r an

swer

to t

he n

eare

st h

un

dre

dth

.

9.5

7 m

r

10-2

Answers (Lesson 10-2)

Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he M

cG

raw

-Hill C

om

pan

ies,

Inc.

Chapter 10 A7 Glencoe Algebra 1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

16

Gle

ncoe A

lgeb

ra 1

En

rich

men

t

Sq

uare

s a

nd

Sq

uare

Ro

ots

Fro

m a

Gra

ph

Th

e g

rap

h o

f y =

x2 c

an

be u

sed

to f

ind

th

e s

qu

are

s an

d s

qu

are

ro

ots

of

nu

mbers

.

To f

ind

th

e s

qu

are

of

3,

loca

te 3

on

th

e x

-axis

. T

hen

fin

d i

ts

corr

esp

on

din

g v

alu

e o

n t

he y

-axis

.

Th

e a

rrow

s sh

ow

th

at

32 =

9.

To f

ind

th

e s

qu

are

root

of

4,

firs

t lo

cate

4 o

n t

he y

-axis

. T

hen

fin

d

its

corr

esp

on

din

g v

alu

e o

n t

he x

-axis

. F

oll

ow

ing t

he a

rrow

s on

th

e

gra

ph

, you

can

see t

hat

√

4 =

2.

A s

mall

part

of

the g

rap

h a

t y =

x2 i

s sh

ow

n b

elo

w.

A 1

:10 r

ati

o

for

un

it l

en

gth

on

th

e y

-axis

to u

nit

len

gth

on

th

e x

-axis

is

use

d.

F

ind

√ �

�

11 .

Th

e a

rrow

s sh

ow

th

at

√

11 =

3.3

to

th

e n

eare

st t

en

th.

Exerc

ises

Use t

he g

ra

ph

ab

ov

e t

o f

ind

ea

ch

of

the f

oll

ow

ing

to

th

e n

ea

rest

wh

ole

nu

mb

er.

1. 1.5

2 2

2. 2.7

2 7

3. 0.9

2 1

4. 3.6

2 13

5. 4.2

2 18

6. 3.9

2 15

Use t

he g

ra

ph

ab

ov

e t

o f

ind

ea

ch

of

the f

oll

ow

ing

to

th

e n

ea

rest

ten

th.

7. √

15 3.9

8. √

8 2.8

9. √

3 1.7

10. √

5 2.2

11. √

14 3.7

12. √

17 4.1

12

1110

20

34

3.3

Ox

y

x

y O

10-2

Exam

ple

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Lesson 10-3

Ch

ap

ter

10

17

Gle

ncoe A

lgeb

ra 1

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Op

era

tio

ns w

ith

Rad

ical

Exp

ressio

ns

Ad

d o

r Su

btr

act

Rad

ical

Exp

ress

ion

s W

hen

ad

din

g o

r su

btr

act

ing r

ad

ical

exp

ress

ion

s, u

se t

he A

ssoci

ati

ve a

nd

Dis

trib

uti

ve P

rop

ert

ies

to s

imp

lify

th

e e

xp

ress

ion

s.

If r

ad

ical

exp

ress

ion

s are

not

in s

imp

lest

form

, si

mp

lify

th

em

.

S

imp

lify

10 √

�

6 -

5 √

�

3 +

6 √

�

3 -

4 √

�

6 .

10 √ #

6 -

5 √ #

3 +

6 √ #

3 -

4 √ #

6 =

(10 -

4) √ #

6 +

(-

5 +

6) √ #

3

Associa

tive a

nd D

istr

ibutive P

ropert

ies

= 6

√ #

6 +

√ #

3

Sim

plif

y.

S

imp

lify

3 √

��

12 +

5 √

��

75 .

3 √ ##

12 +

5 √ ##

75 =

3 √ ###

22 ·

3 +

5 √ ###

52 ·

3

Sim

plif

y.

= 3

· 2

√ #

3 +

5 ·

5 √ #

3

Sim

plif

y.

= 6

√ #

3 +

25 √ #

3

Sim

plif

y.

= 3

1 √ #

3

Dis

trib

utive P

ropert

y

Exerc

ises

Sim

pli

fy e

ach

ex

pressio

n.

1. 2 √ #

5 +

4 √ #

5 6 √

�

5

2. √ #

6 -

4 √ #

6 -

3 √

�

6

3. √ #

8 -

√ #

2 √

�

2

4. 3 √ #

75 +

2 √ #

5 1

5 √

�

3 +

2 √

�

5

5. √ #

20 +

2 √ #

5 -

3 √ #

5 √

�

5

6. 2 √ #

3 +

√ #

6 -

5 √ #

3 -

3 √

�

3 +

√ �

6

7. √ #

12 +

2 √ #

3 -

5 √ #

3 -

√ �

3

8. 3 √ #

6 +

3 √ #

2 -

√ #

50 +

√ #

24 5

√ �

6 -

2 √

�

2

9. √ #

8a -

√ #

2a +

5 √ #

2a 6

√ �

�

2a

10. √ #

54 +

√ #

24 5

√ �

6

11. √ #

3 +

√ # 1 −

3

4

√ �

3

−

3

12. √ #

12 +

√ # 1 −

3

7

√ �

3

−

3

13. √ #

54 -

√ # 1 −

6

1

7 √

�

6

−

6

14. √ #

80 -

√ #

20 +

√ ##

180 8

√ �

5

15. √ #

50 +

√ #

18 -

√ #

75 +

√ #

27 8

√ �

2 -

2 √

�

3

16. 2 √ #

3 -

4 √ #

45 +

2 √ #

1 −

3

8

√ �

3

−

3

- 1

2 √

�

5

17. √ ##

125 -

2 √ #

1 −

5 +

√ # 1 −

3

2

3 √

�

5

−

5

+ √

�

3

−

3

18. √ #

2 −

3 +

3 √ #

3 -

4 √ #

1 −

12

√

�

6 +

7 √

�

3

−

3

10-3

Exam

ple

1

Exam

ple

2

Answers (Lesson 10-2 and Lesson 10-3)

Co

pyrig

ht ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f Th

e M

cG

raw

-Hill C

om

pan

ies, In

c.

Chapter 10 A8 Glencoe Algebra 1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

18

Gle

ncoe A

lgeb

ra 1

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n (c

on

tin

ued

)

Op

era

tio

ns w

ith

Rad

ical

Exp

ressio

ns

Mu

ltip

ly R

ad

ical

Exp

ress

ion

s M

ult

iply

ing t

wo r

ad

ical

exp

ress

ion

s w

ith

dif

fere

nt

rad

ican

ds

is s

imil

ar

to m

ult

iply

ing b

inom

ials

.

M

ult

iply

(3 √

�

2 -

2 √

�

5 )(

4 √

��

20 +

√ �

8 ).

Use

th

e F

OIL

meth

od

.

(3 √

�

2 -

2 √

�

5 )(

4 √

��

20 +

√ �

8 ) =

(3 √

�

2 )(

4 √

��

20 ) +

(3 √

�

2 )(

√ �

8 ) +

(-2 √

�

5 )(

4 √

��

20 ) +

(-2 √

�

5 )(

√ �

8 )

= 1

2 √

��

40 +

3 √

��

16 -

8 √

��

100 -

2 √

��

40

Multip

ly.

= 1

2 √

��

�

22 ·

10 +

3 ·

4 -

8 ·

10 -

2 √

��

�

22 ·

10

Sim

plif

y.

= 2

4 √

��

10 +

12 -

80 -

4 √

��

10

Sim

plif

y.

= 2

0 √

��

10 -

68

Com

bin

e lik

e t

erm

s.

Exerc

ises

Sim

pli

fy e

ach

ex

pressio

n.

1. 2( √

�

3 +

4 √

�

5 )

2

√ �

3 +

8 √

�

5

2. √

�

6 ( √

�

3 -

2 √

�

6 )

3 √

�

2 -

12

3. √

�

5 ( √

�

5 -

√ �

2 )

5 -

√ �

�

10

4. √

�

2 (3

√ �

7 +

2 √

�

5 )

3 √

��

14 +

2 √

��

10

5. (2

- 4

√ �

2 )(

2 +

4 √

�

2 )

-

28

6. (3

+ √

�

6 ) 2

1

5 +

6 √

�

6

7. (2

- 2

√ �

5 ) 2

24 -

8 √

�

5

8. 3 √

�

2 ( √

�

8 +

√ �

24 )

12 +

12

√ �

3

9. √

�

8 ( √

�

2 +

5 √

�

8 )

44

10. ( √

�

5 -

3 √

�

2 )(

√ �

5 +

3 √

�

2 )

-13

11. ( √

�

3 +

√ �

6 ) 2

9 +

6 √

�

2

12. ( √

�

2 -

2 √

�

3 ) 2

1

4 -

4 √

�

6

13. ( √

�

5 -

√ �

2 )(

√ �

2 +

√ �

6 )

14. ( √

�

8 -

√ �

2 )(

√ �

3 +

√ �

6 )

√

��

10 -

2 +

√ ��

30 -

2 √

�

3

√

�

6 +

2 √

�

3

15. ( √

�

5 -

√ �

18 )(

7 √

�

5 +

√ �

3 )

16. (2

√ �

3 -

√ �

45 )(

√ �

12 +

2 √

�

6 )

35 +

√ ��

15 -

21 √

��

10 -

3 √

�

6

12 -

6 √

��

15 +

12 √

�

2 -

6 √

��

30

17. (2

√ �

5 -

2 √

�

3 )(

√ �

10 +

√ �

6 )

18. ( √

�

2 +

3 √

�

3 )(

√ �

12 -

4 √

�

8 )

4 √

�

2

2 -

22 √

�

6

10-3

Exam

ple

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Lesson 10-3

Ch

ap

ter

10

19

Gle

ncoe A

lgeb

ra 1

Sk

ills

Pra

ctic

e

Op

era

tio

ns w

ith

Rad

ical

Exp

ressio

ns

10-3

Sim

pli

fy e

ach

ex

pressio

n.

1. 7 √

�

7 -

2 √

�

7

5 √

�

7

2. 3 √

��

13 +

7 √

��

13

10 √

��

13

3. 6 √

�

5 -

2 √

�

5 +

8 √

�

5 12 √

�

5

4.

√ �

�

15 +

8 √

��

15 -

12

√ �

�

15

-3 √

��

15

5. 12 √

� r -

9 √

� r 3 √

�

r

6. 9 √

��

6a

- 1

1 √

��

6a

+ 4

√ �

�

6a

2 √

��

6a

7. √

��

44 -

√ �

�

11

√ �

�

11

8.

√ �

�

28 +

√ �

�

63

5 √

�

7

9. 4 √

�

3 +

2 √

��

12

8 √

�

3

10. 8 √

��

54 -

4 √

�

6

20 √

�

6

11. √

��

27 +

√ �

�

48 +

√ �

�

12

9 √

�

3

12.

√ �

�

72 +

√ �

�

50 -

√ �

8

9 √

�

2

13. √

��

180 -

5 √

�

5 +

√ �

�

20

3 √

�

5

14. 2 √

��

24 +

4 √

��

54 +

5 √

��

96

36 √

�

6

15. 5 √

�

8 +

2 √

��

20 -

√ �

8

16. 2 √

�

13 +

4 √

�

2 -

5 √

�

13 +

√ �

2

8 √

�

2 +

4 √

�

5

-3 √

��

13 +

5 √

�

2

17.

√ �

2 ( √

�

8 +

√ �

6 )

4 +

2 √

�

3

18.

√ �

5 ( √

�

10 -

√ �

3 )

5 √

�

2 -

√ �

�

15

19.

√ �

6 (3

√ �

2 -

2 √

�

3 )

6 √

�

3 -

6 √

�

2

20. 3 √

�

3 (2

√ �

6 +

4 √

�

10 )

18 √

�

2 +

12 √

��

30

21.

( 4 +

√ �

3 ) (

4 -

√ �

3 )

13

22. (2

- √

�

6 )2

10 -

4 √

�

6

23.

( √ �

8 +

√ �

2 ) (

√ �

5 +

√ �

3 )

24. ( √

�

6 +

4 √

�

5 )(

4 √

�

3 -

√ �

10 )

3 √

��

10 +

3 √

�

6

-8 √

�

2 +

14 √

��

15

Answers (Lesson 10-3)

Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he M

cG

raw

-Hill C

om

pan

ies,

Inc.

Chapter 10 A9 Glencoe Algebra 1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

20

Gle

ncoe A

lgeb

ra 1

Practi

ce

Op

era

tio

ns w

ith

Rad

ical

Exp

ressio

ns

Sim

pli

fy e

ach

ex

pressio

n.

1. 8 √

�

30 -

4 √

�

30 4

√ �

�

30

2. 2 √

�

5 -

7 √

�

5 -

5 √

�

5 -

10

√ �

5

3. 7 √

��

13x -

14 √

��

13x +

2 √

��

13x -

5 √

��

13

x

4. 2 √

��

45 +

4 √

�

20 1

4 √

�

5

5. √

�

40 -

√ �

10 +

√ �

90 4

√ �

�

10

6. 2 √

�

32 +

3 √

�

50 -

3 √

�

18 14

√ �

2

7. √

�

27 +

√ �

18 +

√ �

�

300 3

√ �

2 +

13

√ �

3

8. 5 √

�

8 +

3 √

�

20 -

√ �

32 6

√ �

2 +

6 √

�

5

9. √

�

14 -

√ �

2

−

7

6

√ �

�

14

−

7

10. √

�

50 +

√ �

32 -

√ �

1

−

2

1

7 √

�

2

−

2

11. 5 √

�

19 +

4 √

�

28 -

8 √

�

19 +

√ �

63

12. 3 √

�

10 +

√ �

75 -

2 √

�

40 -

4 √

�

12

-

3 √

��

19 +

11 √

�

7

-

√ �

�

10 -

3 √

�

3

13. √

�

6 ( √

�

10 +

√ �

15 )

2

√ �

�

15 +

3 √

��

10

14. √

�

5 (5

√ �

2 -

4 √

�

8 )

-3

√ �

�

10

15. 2 √

�

7 (3

√ �

12 +

5 √

�

8 )

12

√ �

�

21 +

20

√ �

�

14

16. (5

- √

�

15 ) 2

40 -

10

√ �

�

15

17. ( √

�

10 +

√ �

6 )

( √ �

30 -

√ �

18 )

4 √

�

3

18. ( √

�

8 +

√ �

12 )

( √ �

48 +

√ �

18 )

36 +

14

√ �

6

19. ( √

�

2 +

2 √

�

8 )(

3 √

�

6 -

√ �

5 )

20. (4

√ �

3 -

2 √

�

5 )(

3 √

�

10 +

5 √

�

6 )

3

0 √

�

3 -

5 √

��

10

2 √

��

30 +

30

√ �

2

21. S

OU

ND

T

he s

peed

of

sou

nd

V i

n m

ete

rs p

er

seco

nd

near

Eart

h’s

su

rface

is

giv

en

by

V

= 2

0 √

��

�

t +

273 ,

wh

ere

t i

s th

e s

urf

ace

tem

pera

ture

in

degre

es

Cels

ius.

a

. W

hat

is t

he s

peed

of

sou

nd

near

Eart

h’s

su

rface

at

15°C

an

d a

t 2°C

in

sim

ple

st f

orm

?

240 √

�

2 m

/s,

100 √

��

11 m

/s

b

. H

ow

mu

ch f

ast

er

is t

he s

peed

of

sou

nd

at

15°C

th

an

at

2°C

?

240 √

�

2 -

100 √

��

11 ≈

7.7

5 m

/s

22. G

EO

ME

TR

Y A

rect

an

gle

is

5 √

�

7 +

2 √

�

3 m

ete

rs l

on

g a

nd

6 √

�

7 -

3 √

�

3 m

ete

rs w

ide.

a

. F

ind

th

e p

eri

mete

r of

the r

ect

an

gle

in

sim

ple

st f

orm

. 22 √

�

7 -

2 √

�

3 m

b

. F

ind

th

e a

rea o

f th

e r

ect

an

gle

in

sim

ple

st f

orm

. 190 -

3 √

��

21 m

2

10-3

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Lesson 10-3

Ch

ap

ter

10

21

Gle

ncoe A

lgeb

ra 1

-5

-1

0

5

5

10

10

15

20

Ski slo

pe

y

xO

Wo

rd

Pro

ble

m P

racti

ce

Op

era

tio

ns w

ith

Rad

ical

Exp

ressio

ns

1. A

RC

HIT

EC

TU

RE

T

he P

en

tagon

is

the

bu

ild

ing t

hat

hou

ses

the U

.S.

Dep

art

men

t of

Defe

nse

. F

ind

th

e

ap

pro

xim

ate

peri

mete

r of

the b

uil

din

g,

wh

ich

is

a r

egu

lar

pen

tagon

. L

eave y

ou

r an

swer

as

a r

ad

ical

exp

ress

ion

.

115 √

��

149 m

2. E

AR

TH

T

he s

urf

ace

are

a o

f a s

ph

ere

w

ith

rad

ius

r i

s giv

en

by t

he f

orm

ula

4π

r2.

Ass

um

ing t

hat

Eart

h i

s cl

ose

to

sph

eri

cal

in s

hap

e a

nd

has

a s

urf

ace

are

a

of

abou

t 5.1

× 1

08 s

qu

are

kil

om

ete

rs,

wh

at

is t

he r

ad

ius

of

Eart

h t

o t

he

neare

st t

en

kil

om

ete

rs?

6370 k

m

3. G

EO

ME

TR

Y T

he a

rea o

f a t

rap

ezoid

is

fou

nd

by m

ult

iply

ing i

ts h

eig

ht

by t

he

avera

ge l

en

gth

of

its

base

s. F

ind

th

e a

rea

of

deck

att

ach

ed

to M

r. W

ilso

n’s

hou

se.

Giv

e y

ou

r an

swer

as

a s

imp

lifi

ed

rad

ical

exp

ress

ion

.

6

3 √ �

�

15 ft

2

4. R

EC

RE

AT

ION

C

arm

en

su

rveyed

a s

ki

slop

e u

sin

g a

dig

ital

devic

e c

on

nect

ed

to

a c

om

pu

ter.

Th

e c

om

pu

ter

mod

el

ass

ign

ed

coord

inate

s to

th

e t

op

an

d

bott

om

poin

ts o

f th

e h

ill

as

show

n i

n t

he

dia

gra

m.

Wri

te a

sim

pli

fied

rad

ical

exp

ress

ion

th

at

rep

rese

nts

th

e s

lop

e o

f th

e h

ill.

5. FR

EE

FA

LL

S

up

pose

a b

all

is

dro

pp

ed

fr

om

a b

uil

din

g w

ind

ow

800 f

eet

in t

he

air

. A

noth

er

ball

is

dro

pp

ed

fro

m a

low

er

win

dow

288 f

eet

hig

h.

Both

ball

s are

re

lease

d a

t th

e s

am

e t

ime.

Ass

um

e a

ir

resi

stan

ce i

s n

ot

a f

act

or

an

d u

se t

he

foll

ow

ing f

orm

ula

to f

ind

how

man

y

seco

nd

s t

it w

ill

tak

e a

ball

to f

all

h f

eet.

t

= 1

−

4

√ �

h

a.

How

mu

ch t

ime w

ill

pass

betw

een

w

hen

th

e f

irst

ball

hit

s th

e g

rou

nd

an

d w

hen

th

e s

eco

nd

ball

hit

s th

e

gro

un

d?

Giv

e y

ou

r an

swer

as

a

sim

pli

fied

rad

ical

exp

ress

ion

.

b.

Wh

ich

ball

lan

ds

firs

t? T

he b

all

dro

pp

ed

fro

m 2

88 f

eet

lan

ds

firs

t.

c.

Fin

d a

deci

mal

ap

pro

xim

ati

on

of

the

an

swer

for

part

a.

Rou

nd

you

r an

swer

to t

he n

eare

st t

en

th.

( 7r+

12

) ˚

Ho

use

De

ck

h =

7 √

� 5 ft

12 √

�

3 f

t

6 √

�

3 f

t

A (

2 √

��

14 ,

5 √

�

7 )

23 √

��

149 m

B (-

2 √

��

14 ,

7 √

�

7 )

10-3

ab

ou

t 2.8

s

2 √

�

2 s

Answers (Lesson 10-3)

Co

pyrig

ht ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f Th

e M

cG

raw

-Hill C

om

pan

ies, In

c.

Chapter 10 A10 Glencoe Algebra 1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

22

Gle

ncoe A

lgeb

ra 1

En

rich

men

t

Th

e W

heel

of T

heo

do

ru

s

Th

e G

reek

math

em

ati

cian

s w

ere

in

trig

ued

by p

roble

ms

of

re

pre

sen

tin

g d

iffe

ren

t n

um

bers

an

d e

xp

ress

ion

s u

sin

g

geom

etr

ic c

on

stru

ctio

ns.

Th

eod

oru

s, a

Gre

ek

ph

iloso

ph

er

wh

o l

ived

abou

t 425 B

.C., i

s sa

id t

o h

ave d

isco

vere

d a

way t

o c

on

stru

ct t

he s

equ

en

ce

√ �

1 ,

√ �

2 ,

√ �

3 ,

√ �

4 ,

. .

..

Th

e b

egin

nin

g o

f h

is c

on

stru

ctio

n i

s sh

ow

n.

You

sta

rt w

ith

an

is

osc

ele

s ri

gh

t tr

ian

gle

wit

h s

ides

1 u

nit

lon

g.

Use t

he f

igu

re a

bo

ve.

Writ

e e

ach

len

gth

as a

ra

dic

al

ex

pressio

n i

n s

imp

lest

form

.

1. li

ne s

egm

en

t A

O √

�

1

2.

lin

e s

egm

en

t B

O

√ �

2

3. li

ne s

egm

en

t C

O

√ �

3

4.

lin

e s

egm

en

t D

O

√ �

4

5. D

esc

ribe h

ow

each

new

tri

an

gle

is

ad

ded

to t

he f

igu

re.

Dra

w a

new

sid

e o

f le

ng

th

1 a

t ri

gh

t an

gle

s t

o t

he l

ast

hyp

ote

nu

se.

Th

en

dra

w t

he n

ew

hyp

ote

nu

se.

6. T

he l

en

gth

of

the h

yp

ote

nu

se o

f th

e f

irst

tri

an

gle

is

√ �

2 .

For

the s

eco

nd

tri

an

gle

, th

e

len

gth

is

√ �

3 .

Wri

te a

n e

xp

ress

ion

for

the l

en

gth

of

the h

yp

ote

nu

se o

f th

e n

th t

rian

gle

.

√ �

��

n +

1

7. S

how

th

at

the m

eth

od

of

con

stru

ctio

n w

ill

alw

ays

pro

du

ce t

he n

ext

nu

mber

in t

he

sequ

en

ce.

(Hin

t: F

ind

an

exp

ress

ion

for

the h

yp

ote

nu

se o

f th

e (

n +

1)t

h t

rian

gle

.)

√ �

��

��

( √ �

n

)2 +

(1)2

or

√ �

��

n +

1

8. In

th

e s

pace

belo

w,

con

stru

ct a

Wh

eel

of

Th

eod

oru

s. S

tart

wit

h a

lin

e s

egm

en

t 1 c

en

tim

ete

r lo

ng.

Wh

en

does

the W

heel

start

to o

verl

ap

?

aft

er

len

gth

√ �

�

18

1

1

1

1

OAB

CD

10-3

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Lesson 10-4

Ch

ap

ter

10

23

Gle

ncoe A

lgeb

ra 1

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Rad

ical

Eq

uati

on

s

Rad

ical

Eq

uati

on

s E

qu

ati

on

s co

nta

inin

g r

ad

icals

wit

h v

ari

able

s in

th

e r

ad

ican

d a

re

call

ed

ra

dic

al

eq

ua

tio

ns.

Th

ese

can

be s

olv

ed

by f

irst

usi

ng t

he f

oll

ow

ing s

tep

s.

S

olv

e 1

6 =

√ �

x

−

2

fo

r x

.

16 =

√ �

x

−

2

O

rigin

al equation

2(1

6)

= 2( √

�

x

−

2 )

Multip

ly e

ach s

ide b

y 2

.

32 =

√ �

x

Sim

plif

y.

(3

2)2

= (

√ �

x )

2

Square

each s

ide.

1024 =

x

Sim

plif

y.

Th

e s

olu

tion

is

1024,

wh

ich

ch

eck

s in

th

e o

rigin

al

equ

ati

on

.

S

olv

e √

��

�

4x -

7 +

2 =

7.

√

��

�

4x -

7 +

2 =

7

Origin

al equation

√ �

��

4x -

7 +

2 -

2 =

7 -

2

Subtr

act

2 f

rom

each s

ide.

√

��

�

4x -

7 =

5

Sim

plif

y.

( √

��

�

4x -

7 )2

= 5

2

Square

each s

ide.

4

x -

7 =

25

Sim

plif

y.

4x -

7 +

7 =

25 +

7

Add 7

to e

ach s

ide.

4

x =

32

Sim

plif

y.

x =

8

Div

ide e

ach s

ide b

y 4

.

Th

e s

olu

tion

is

8,

wh

ich

ch

eck

s in

th

e o

rigin

al

equ

ati

on

.

Exerc

ises

So

lve e

ach

eq

ua

tio

n.

Ch

eck

yo

ur s

olu

tio

n.

1. √

�

a =

8

64

2. √

�

a +

6 =

32 676

3. 2 √

�

x =

8 16

4. 7 =

√ �

��

26 -

n

-23

5. √

��

-a

= 6

-

36

6. √

��

3r2

= 3

±

√ �

3 ±

√ �

7. 2 √

�

3 =

√ �

y 12

8. 2 √

��

3a

- 2

= 7

6 3

−

4

9. √

��

�

x -

4 =

6 40

10. √

��

�

2m

+ 3

= 5

11

11. √

��

�

3b

- 2

+ 1

9 =

24 9

12. √

��

�

4x -

1 =

3 5

−

2

13. √

��

�

3r

+ 2

= 2

√ �

3

10

−

3

14. √ �

x

−

2 =

1

−

2

1

−

2

15. √ �

x

−

8 =

4 128

16. √

��

��

6x

2 +

5x =

2 1

−

2 ,

- 4

−

3

17. √ �

x

−

3 +

6 =

8 12

18. 2 √ �

�

3x

−

5

+ 3

= 1

1 26 2

−

3

10-4

Exam

ple

1Exam

ple

2

Ste

p 1

Is

ola

te t

he r

ad

ical

on

on

e s

ide o

f th

e e

qu

ati

on

.

Ste

p 2

S

qu

are

each

sid

e o

f th

e e

qu

ati

on

to e

lim

inate

th

e r

ad

ical.

Answers (Lesson 10-3 and Lesson 10-4)

Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he M

cG

raw

-Hill C

om

pan

ies,

Inc.

Chapter 10 A11 Glencoe Algebra 1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

24

Gle

ncoe A

lgeb

ra 1

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n(c

on

tin

ued

)

Rad

ical

Eq

uati

on

s

Extran

eo

us S

olu

tio

ns

To s

olv

e a

rad

ical

equ

ati

on

wit

h a

vari

able

on

both

sid

es,

you

n

eed

to s

qu

are

each

sid

e o

f th

e e

qu

ati

on

. S

qu

ari

ng e

ach

sid

e o

f an

equ

ati

on

som

eti

mes

pro

du

ces

ex

tra

neo

us s

olu

tio

ns,

or

solu

tion

s th

at

are

not

solu

tion

s of

the o

rigin

al

equ

ati

on

. T

here

fore

, it

is

very

im

port

an

t th

at

you

ch

eck

each

solu

tion

.

S

olv

e √

��

�

x +

3 =

x -

3.

√

��

�

x +

3 =

x -

3

Origin

al equation

( √ �

��

x +

3 )2

= (

x -

3)2

S

quare

each s

ide.

x +

3 =

x2 -

6x +

9

Sim

plif

y.

0 =

x2 -

7x +

6

Subtr

act

x a

nd 3

fro

m e

ach s

ide.

0 =

(x -

1)(

x -

6)

Facto

r.

x -

1 =

0 or

x -

6 =

0

Zero

Pro

duct

Pro

pert

y

x =

1

x =

6

Solv

e.

CH

EC

K √

��

�

x +

3

= x

- 3

√

��

�

x +

3 =

x -

3

√

��

�

1 +

3

! 1

- 3

√

��

�

6 +

3 !

6 -

3

√ �

4 !

-2

√ �

9 !

3

2 ≠

-2

3 =

3 #

Sin

ce x

= 1

does

not

sati

sfy t

he o

rigin

al

equ

ati

on

, x =

6 i

s th

e o

nly

solu

tion

.

Exerc

ises

So

lve e

ach

eq

ua

tio

n.

Ch

eck

yo

ur s

olu

tio

n.

1. √

�

a =

a 0,

1

2. √

��

�

a +

6 =

a 3

3. 2 √

�

x =

x 0,

4

4. n

= √

��

�

2 -

n 1

5. √

��

-a

= a

0

6. √

��

��

10 -

6k +

3 =

k

!

7. √

��

�

y -

1 =

y -

1 1,

2

8. √

��

�

3a

- 2

= a

1,

2

9. √

��

�

x +

2 =

x 2

10. √

��

�

2b

+ 5

= b

- 5

10

11. √

��

�

3b

+ 6

= b

+ 2

1

12. √

��

�

4x -

4 =

x 2

13. r

+ √

��

�

2 -

r =

2 1,

2

14. √

��

��

x2 +

10x =

x +

4 8

15.

-2 √ �

x

−

8 =

15

!

16. √

��

��

6x

2 -

4x =

x +

2

17. √

��

��

2y

2 -

64

= y

18. √

��

��

��

3

x2 +

12

x +

1 =

x +

5

- 2

−

5 ,

2

8

-4,

3

10-4

Exam

ple

1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Lesson 10-4

Ch

ap

ter

10

25

Gle

ncoe A

lgeb

ra 1

Sk

ills

Pra

ctic

e

Rad

ical

Eq

uati

on

s

So

lve e

ach

eq

ua

tio

n.

Ch

eck

yo

ur s

olu

tio

n.

1. √

� f =

7 49

2. √

��

-x =

5

-25

3. √

��

5p

= 1

0 20

4. √

�

4y =

6 9

5. 2 √

�

2 =

√ �

u

8

6. 3 √

�

5 =

√ �

�

-n

-

45

7. √

�

g -

6 =

3 81

8. √

��

5a

+ 2

= 0

ø

9. √

��

�

2t

- 1

= 5

13

10. √

��

�

3k -

2 =

4 6

11. √

��

�

x +

4 -

2 =

1 5

12. √

��

�

4x -

4 -

4 =

0 5

13. √

�

d

−

3 =

4 144

14. √ �

m

−

3

= 3

27

15. x =

√ �

��

x +

2 2

16. d

= √

��

�

12 -

d 3

17. √

��

�

6x -

9 =

x 3

18. √

��

�

6p

- 8

= p

2,

4

19. √

��

�

x +

5 =

x -

1 4

20. √

��

�

8 -

d =

d -

8 8

21. √

��

�

r -

3 +

5 =

r 7

22. √

��

�

y -

1 +

3 =

y 5

23. √

��

�

5n

+ 4

= n

+ 2

1,

0

24. √

��

�

3z

- 6

= z

- 2

5,

2

10-4

Answers (Lesson 10-4)

Co

pyrig

ht ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f Th

e M

cG

raw

-Hill C

om

pan

ies, In

c.

Chapter 10 A12 Glencoe Algebra 1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

26

Gle

ncoe A

lgeb

ra 1

Practi

ce

Rad

ical

Eq

uati

on

sS

olv

e e

ach

eq

ua

tio

n.

Ch

eck

yo

ur s

olu

tio

n.

1. √

��

-b =

8 -

64

2

. 4 √

�

3 =

√ � x

48

3. 2 √

�

4r +

3 =

11 4

4. 6 -

√ �

2

y =

-2 32

5. √

��

�

k +

2 -

3 =

7 98

6. √

��

�

m -

5 =

4 √

�

3 53

7. √

��

�

6t

+ 1

2 =

8 √

�

6 62

8. √

��

�

3j

- 1

1 +

2 =

9 20

9. √

��

��

2x +

15

+ 5

= 1

8 77

10. √ �

3

d

−

5

- 4

= 2

60

11. 6 √ �

�

3x

−

3

- 3

= 0

1

−

4

12. 6 +

√ �

5

r −

6

= -

2 "

13. y

= √

��

�

y +

6 3

14. √

��

��

15 -

2x =

x 3

15. √

��

�

w +

4 =

w +

4 -

4, -

3

16. √

��

�

17 -

k =

k -

5 8

17. √

��

��

5m

- 1

6 =

m -

2 4,

5

18. √

��

��

24 +

8q =

q +

3 -

3,

5

19. √

��

��

4t

+ 1

7 -

t -

3 =

0 2

20. 4 -

√ �

��

�

3m

+ 2

8 =

m -

1

21. √

��

��

10

p +

61

- 7

= p

-

6,

2

22. √

��

�

2x

2 -

9 =

x 3

23. E

LE

CT

RIC

ITY

T

he v

olt

age V

in

a c

ircu

it i

s giv

en

by V

= √

��

PR

, w

here

P i

s th

e p

ow

er

in

watt

s an

d R

is

the r

esi

stan

ce i

n o

hm

s.

a

. If

th

e v

olt

age i

n a

cir

cuit

is

120 v

olt

s an

d t

he c

ircu

it p

rod

uce

s 1500 w

att

s of

pow

er,

w

hat

is t

he r

esi

stan

ce i

n t

he c

ircu

it?

b

. S

up

pose

an

ele

ctri

cian

desi

gn

s a c

ircu

it w

ith

110 v

olt

s an

d a

resi

stan

ce o

f 10 o

hm

s.

How

mu

ch p

ow

er

wil

l th

e c

ircu

it p

rod

uce

?

24. FR

EE

FA

LL

A

ssu

min

g n

o a

ir r

esi

stan

ce,

the t

ime t

in

seco

nd

s th

at

it t

ak

es

an

obje

ct t

o

fall

h f

eet

can

be d

ete

rmin

ed

by t

he e

qu

ati

on

t =

√ �

h

−

4 .

a

. If

a s

kyd

iver

jum

ps

from

an

air

pla

ne a

nd

fre

e f

all

s fo

r 10 s

eco

nd

s befo

re o

pen

ing t

he

para

chu

te,

how

man

y f

eet

does

the s

kyd

iver

fall

?

b

. S

up

pose

a s

eco

nd

sk

yd

iver

jum

ps

an

d f

ree f

all

s fo

r 6 s

eco

nd

s. H

ow

man

y f

eet

does

the

seco

nd

sk

yd

iver

fall

?

10-4

1600 f

t

9.6

oh

ms

1210 w

att

s

576 f

t

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Lesson 10-4

Ch

ap

ter

10

27

Gle

ncoe A

lgeb

ra 1

Wo

rd

Pro

ble

m P

racti

ce

Rad

ical

Eq

uati

on

s

1. S

UB

MA

RIN

ES

T

he d

ista

nce

in

mil

es

that

the l

ook

ou

t of

a s

ubm

ari

ne c

an

see

is a

pp

roxim

ate

ly d

= 1

.22 √

�

h ,

wh

ere

h i

s th

e h

eig

ht

in f

eet

above t

he s

urf

ace

of

the w

ate

r. H

ow

far

wou

ld a

su

bm

ari

ne

peri

scop

e h

ave t

o b

e a

bove t

he w

ate

r to

lo

cate

a s

hip

6 m

iles

aw

ay?

Rou

nd

you

r an

swer

to t

he n

eare

st t

en

th.

2. P

ET

S F

ind

th

e v

alu

e o

f x i

f th

e p

eri

mete

r of

a t

rian

gu

lar

dog p

en

is

25 m

ete

rs.

x =

8

3. LO

GG

ING

D

oyle

’s l

og r

ule

est

imate

s th

e

am

ou

nt

of

usa

ble

lu

mber

(in

board

feet)

th

at

can

be m

ille

d f

rom

a s

hip

men

t of

logs.

It

is r

ep

rese

nte

d b

y t

he e

qu

ati

on

B =

L ( d

- 4

−

4

) 2

, w

here

d i

s th

e l

og

d

iam

ete

r (i

n i

nch

es)

an

d L

is

the l

og

len

gth

(in

feet)

. S

up

pose

th

e t

ruck

ca

rrie

s 20 l

ogs,

each

25 f

eet

lon

g,

an

d

that

the s

hip

men

t yie

lds

a t

ota

l of

6000

board

feet

of

lum

ber.

Est

imate

th

e

dia

mete

r of

the l

ogs

to t

he n

eare

st i

nch

. A

ssu

me t

hat

all

th

e l

ogs

have u

nif

orm

le

ngth

an

d d

iam

ete

r.

18 i

n.

4. FIR

EFIG

HT

ING

F

ire f

igh

ters

calc

ula

te

the f

low

rate

of

wate

r ou

t of

a p

art

icu

lar

hyd

ran

t by u

sin

g t

he f

oll

ow

ing f

orm

ula

.

F =

26.9

d2 √

�

p

F

is

the f

low

rate

(in

gall

on

s p

er

min

ute

),

p i

s th

e n

ozzle

pre

ssu

re (

in p

ou

nd

s p

er

squ

are

in

ch),

an

d d

is

the d

iam

ete

r of

the

hose

(in

in

ches)

. In

ord

er

to e

ffect

ively

fi

gh

t a f

ire,

the c

om

bin

ed

flo

w r

ate

of

two

hose

s n

eed

s to

be a

bou

t 2430 g

all

on

s p

er

min

ute

. T

he d

iam

ete

r of

each

of

the

hose

s is

3 i

nch

es,

bu

t th

e n

ozzle

pre

ssu

re

of

on

e h

ose

is

4 t

imes

that

of

the s

eco

nd

h

ose

. W

hat

are

th

e n

ozzle

pre

ssu

res

for

each

hose

? R

ou

nd

you

r an

swers

to t

he

neare

st t

en

th.

11.2

psi

an

d 4

4.8

psi

5. G

EO

ME

TR

Y T

he l

ate

ral

surf

ace

are

a s

of

a r

igh

t ci

rcu

lar

con

e,

not

incl

ud

ing t

he

base

, is

rep

rese

nte

d b

y t

he e

qu

ati

on

s =

πr

√ �

��

r2 +

h2 ,

wh

ere

r i

s th

e r

ad

ius

of

the c

ircu

lar

base

an

d h

is

the h

eig

ht

of

the c

on

e.

a.

If t

he l

ate

ral

surf

ace

are

a o

f a f

un

nel

is 1

27.5

4 s

qu

are

cen

tim

ete

rs a

nd

its

ra

diu

s is

3.5

cen

tim

ete

rs,

fin

d i

ts

heig

ht

to t

he n

eare

st t

en

th o

f a

cen

tim

ete

r.

b.

Wh

at

is t

he a

rea o

f th

e o

pen

ing (

i.e.,

the b

ase

) of

the f

un

nel?

x+ 1

m

12

m 1

0 m

10-4

11.1

cm

38.5

cm

2

24.2

ft

Answers (Lesson 10-4)

Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he M

cG

raw

-Hill C

om

pan

ies,

Inc.

Chapter 10 A13 Glencoe Algebra 1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

28

Gle

ncoe A

lgeb

ra 1

En

rich

men

t

Mo

re T

han

On

e S

qu

are

Ro

ot

You

have l

earn

ed

th

at

to r

em

ove t

he s

qu

are

root

in a

n e

qu

ati

on

, you

fir

st n

eed

to i

sola

te t

he

squ

are

root,

th

en

squ

are

both

sid

es

of

the e

qu

ati

on

, an

d f

inall

y,

solv

e t

he r

esu

ltin

g e

qu

ati

on

. H

ow

ever,

th

ere

are

equ

ati

on

s th

at

con

tain

more

th

at

on

e s

qu

are

root

an

d s

imp

ly s

qu

ari

ng

on

ce i

s n

ot

en

ou

gh

to r

em

ove a

ll o

f th

e r

ad

icals

.

S

olv

e √

��

�

x +

7 =

√ �

x

+ 1

.

√

��

�

x +

7 =

√ �

x +

1

One o

f th

e s

quare

roots

is a

lready isola

ted.

( √ �

��

x +

7 ) 2

= (

√ �

x +

1 ) 2

S

quare

both

sid

es t

o r

em

ove t

he s

quare

root.

x

+ 7

= x

+ 2

√ �

x +

1

Sim

plif

y.

Use t

he F

OIL

meth

od t

o s

quare

rig

ht

sid

e.

x +

7 -

x -

1 =

2 √

�

x

Sim

plif

y.

6 =

2 √

�

x

Sim

plif

y.

Isola

te t

he s

quare

root

term

again

.

3 =

√ �

x

Div

ide b

oth

sid

es b

y 2

.

9 =

x

Square

both

sid

es t

o r

em

ove t

he s

quare

root.

Ch

eck

: S

ubst

itu

te i

nto

th

e o

rigin

al

equ

ati

on

to m

ak

e s

ure

you

r so

luti

on

is

vali

d.

√ �

��

9 +

7 =

√ �

9 +

1

Repla

ce x

with 9

.

√

��

16 =

3 +

1

S

impify.

4 =

4 !

The e

quation is t

rue,

so x

= 9

is t

he s

olu

tion.

Exerc

ises

So

lve e

ach

eq

ua

tio

n.

1.

√ �

��

x +

13 -

2 =

√ �

��

x +

1

3

2.

√ �

��

x +

11 =

√ �

��

x +

3 +

2

-2

3.

√ �

��

x +

9 -

3 =

√ �

��

x -

6

7

4.

√ �

��

x +

21 =

√ �

x +

3 4

5.

√ �

��

x +

9 +

3 =

√ �

��

x +

20 +

2 16

6.

√ �

��

x -

6 +

6 =

√ �

��

x +

1 +

5 15

10-4

Exam

ple

Lesson 10-4

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

29

Gle

ncoe A

lgeb

ra 1

Gra

ph

ing C

alc

ula

tor

Act

ivit

y

Rad

ical

Ineq

ualiti

es

Th

e g

rap

hs

of

rad

ical

equ

ati

on

s ca

n b

e u

sed

to d

ete

rmin

e t

he s

olu

tion

s of

rad

ical

inequ

ali

ties

thro

ugh

th

e C

AL

C m

en

u.

S

olv

e e

ach

in

eq

ua

lity

.

a.

√ �

��

x +

4 ≤

3

En

ter

√ �

��

x +

4 i

n Y

1 a

nd

3 i

n Y

2 a

nd

gra

ph

. E

xam

ine t

he g

rap

hs.

Use

TR

AC

E t

o f

ind

th

e e

nd

poin

t of

the

gra

ph

of

the r

ad

ical

equ

ati

on

. U

se

CA

LC

to d

ete

rmin

e t

he i

nte

rsect

ion

of

the g

rap

hs.

Th

is i

nte

rval,

-4 t

o 5

,

wh

ere

th

e g

rap

h o

f y =

√ �

��

x +

4

is

belo

w t

he g

rap

h o

f y =

3,

rep

rese

nts

the s

olu

tion

to t

he i

nequ

ali

ty.

Th

us,

the s

olu

tion

is

-4 ≤

x ≤

5.

b.

√ �

��

2x

- 5

> x

- 4

Gra

ph

each

sid

e o

f th

e i

nequ

ali

ty.

Fin

d

the i

nte

rsect

ion

an

d t

race

to t

he e

nd

poin

t of

the r

ad

ical

gra

ph

.

Th

e g

rap

h o

f y =

√ �

��

2x -

5 i

s above t

he

gra

ph

of

y =

x -

4 f

rom

2.5

up

to 7

. T

hu

s,

the s

olu

tion

is

2.5

< x

< 7

.

Exerc

ises

So

lve e

ach

in

eq

ua

lity

.

1. 6 -

√ �

��

2x +

1 <

3

2.

√ �

��

4x -

5 ≤

7

3.

√ �

��

5x -

4 ≥

4

x

> 4

5

−

4 ≤

x ≤

27

−

2

x ≥

4

4.

-4 >

√ �

��

3x -

2

5.

√ �

��

3x -

6 +

5 ≥

-3

6.

√ �

��

6 -

3x <

x +

16

n

o s

olu

tio

n

x

≥ 2

-10

< x

< 2

[-1

0,

10

] sc

l:1

by

[-1

0,

10

] sc

l:1

[-1

0,

10

] sc

l:1

by

[-1

0,

10

] sc

l:1

[-1

0,

10

] sc

l:1

by

[-1

0,

10

] sc

l:1

[-1

0,

10

] sc

l:1

by

[-1

0,

10

] sc

l:1

10-4

Exam

ple

Answers (Lesson 10-4)

Co

pyrig

ht ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f Th

e M

cG

raw

-Hill C

om

pan

ies, In

c.

Chapter 10 A14 Glencoe Algebra 1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

30

Gle

ncoe A

lgeb

ra 1

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Th

e P

yth

ag

ore

an

Th

eo

rem

Th

e P

yth

ag

ore

an

Th

eo

rem

T

he s

ide o

pp

osi

te t

he r

igh

t an

gle

in

a r

igh

t tr

ian

gle

is

call

ed

th

e h

yp

ote

nu

se

. T

his

sid

e i

s alw

ays

the l

on

gest

sid

e o

f a r

igh

t tr

ian

gle

. T

he o

ther

two s

ides

are

call

ed

th

e l

eg

s o

f th

e t

rian

gle

. T

o f

ind

th

e l

en

gth

of

an

y s

ide o

f a r

igh

t

tria

ngle

, giv

en

th

e l

en

gth

s of

the o

ther

two s

ides,

you

can

use

th

e P

yth

ag

orea

n T

heo

rem

.

Pyth

ag

ore

an

Th

eo

rem

If a

and b

are

the m

easure

s o

f th

e legs o

f a r

ight

tria

ngle

and c

is t

he m

easure

of

the h

ypote

nuse,

then c

2 =

a2 +

b2.

CB

Ab

ac

F

ind

th

e l

en

gth

of

the m

issin

g s

ide.

c2 =

a2 +

b2

Pyth

agore

an T

heore

m

c2 =

52 +

12

2

a =

5 a

nd b

= 1

2

c2 =

169

Sim

plif

y.

c =

√ ""

169

Take t

he s

quare

root

of

each s

ide.

c =

13

Th

e l

en

gth

of

the h

yp

ote

nu

se i

s 13.

Exerc

ises

Fin

d t

he l

en

gth

of

ea

ch

mis

sin

g s

ide.

If n

ecessa

ry

, ro

un

d t

o t

he

nea

rest

hu

nd

red

th.

1.

2.

3.

5

0

45.8

3

35.3

625

25

c

100

110

a

40

30

c

10-5

12

5

14

8

415

89

5

Exam

ple

4.

5.

6.

16.1

2

5.5

7

8

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Lesson 10-5

Ch

ap

ter

10

31

Gle

ncoe A

lgeb

ra 1

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n(c

on

tin

ued

)

Th

e P

yth

ag

ore

an

Th

eo

rem

Rig

ht

Tri

an

gle

s If

a a

nd

b a

re t

he m

easu

res

of

the s

hort

er

sid

es

of

a t

rian

gle

, c i

s th

e

measu

re o

f th

e l

on

gest

sid

e,

an

d c

2 =

a2 +

b2,

then

th

e t

rian

gle

is

a r

igh

t tr

ian

gle

.

D

ete

rm

ine w

heth

er t

he f

oll

ow

ing

sid

e m

ea

su

res f

orm

rig

ht

tria

ng

les.

a.

10,

12,

14

Sin

ce t

he m

easu

re o

f th

e l

on

gest

sid

e i

s 14,

let

c =

14,

a =

10,

an

d b

= 1

2.

c

2 =

a2 +

b2

Pyth

agore

an T

heore

m

14

2 #

10

2 +

12

2

a =

10,

b =

12,

c =

14

196 #

100 +

144

Multip

ly.

196 ≠

244

Add.

Sin

ce c

2 ≠

a2 +

b2,

the t

rian

gle

is

not

a r

igh

t tr

ian

gle

.

b.

7,

24,

25

Sin

ce t

he m

easu

re o

f th

e l

on

gest

sid

e i

s 25,

let

c =

25,

a =

7,

an

d b

= 2

4.

c

2 =

a2 +

b2

Pyth

agore

an T

heore

m

25

2 #

72 +

24

2

a =

7,

b =

24,

c =

25

625 #

49

+ 5

76

Multip

ly.

625 =

625

Add.

Sin

ce c

2 =

a2 +

b2,

the t

rian

gle

is

a r

igh

t tr

ian

gle

.

Exerc

ises

Dete

rm

ine w

heth

er e

ach

set

of

mea

su

res c

an

be s

ides o

f a

rig

ht

tria

ng

le.

Th

en

dete

rm

ine w

heth

er t

hey

fo

rm

a P

yth

ag

orea

n t

rip

le.

1. 14,

48,

50 yes;

yes

2. 6,

8,

10 yes;

yes

3. 8,

8,

10 n

o;

no

4. 90,

120,

150 yes;

yes

5. 15,

20,

25 yes;

yes

6. 4,

8,

4 √ "

5 yes;

no

7. 2,

2,

√ "

8 yes;

no

8. 4,

4,

√ ""

20 n

o;

no

9. 25,

30,

35 n

o;

no

10. 24,

36,

48 n

o;

no

11. 18,

80,

82 yes;

yes

12. 150,

200,

250 yes;

yes

13. 100,

200,

300 n

o;

no

14. 500,

1200,

1300 yes;

yes

15. 700,

1000,

1300 n

o;

no

10-5

Exam

ple

Answers (Lesson 10-5)

Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he M

cG

raw

-Hill C

om

pan

ies,

Inc.

Chapter 10 A15 Glencoe Algebra 1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

32

Gle

ncoe A

lgeb

ra 1

Sk

ills

Practi

ce

Th

e P

yth

ag

ore

an

Th

eo

rem

Fin

d t

he l

en

gth

of

ea

ch

mis

sin

g s

ide.

If n

ecessa

ry

, ro

un

d t

o t

he n

ea

rest

hu

nd

red

th.

1.

2.

75

36

3.

4.

30

15.7

5

5.

6.

9.8

5

70

Dete

rm

ine w

heth

er e

ach

set

of

mea

su

res c

an

be s

ides o

f a

rig

ht

tria

ng

le.

Th

en

dete

rm

ine w

heth

er t

hey

fo

rm

a P

yth

ag

orea

n t

rip

le.

7.

7,

24,

25 yes;

yes

8.

15,

30,

34 n

o;

no

9.

16,

28,

32 n

o;

no

10.

18,

24,

30 yes;

yes

11.

15,

36,

39 yes;

yes

12.

5,

7,

√ �

�

74 yes;

no

13.

4,

5,

6 n

o;

no

14.

10,

11,

√ �

�

221 yes;

no

250240

a4

9

c

29

33

b

16

34

b

15

39

a

21

72

c

10-5

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Lesson 10-5

Ch

ap

ter

10

33

Gle

ncoe A

lgeb

ra 1

Practi

ce

Th

e P

yth

ag

ore

an

Th

eo

rem

Fin

d t

he l

en

gth

of

ea

ch

mis

sin

g s

ide.

If n

ecessa

ry

, ro

un

d t

o t

he

nea

rest

hu

nd

red

th.

1.

2.

3.

68

15.4

9

11.3

1

Dete

rm

ine w

heth

er e

ach

set

of

mea

su

res c

an

be s

ides o

f rig

ht

tria

ng

le.

Th

en

dete

rm

ine w

heth

er t

hey

fo

rm

a P

yth

ag

orea

n t

rip

le.

4. 11,

18,

21

5. 21,

72,

75

n

o;

no

y

es;

yes

6. 7,

8,

11

7. 9,

10,

√ �

�

161

n

o;

yes

no

; n

o

8. 9,

2 √

��

10 ,

11

9.

√ �

7 ,

2 √

�

2 ,

√ �

�

15

y

es;

no

yes;

no

10. S

TO

RA

GE

T

he s

hed

in

Ste

ph

an

’s b

ack

yard

has

a d

oor

that

measu

res

6 f

eet

hig

h a

nd

3 f

eet

wid

e.

Ste

ph

an

wou

ld l

ike t

o s

tore

a s

qu

are

th

eate

r p

rop

th

at

is 7

feet

on

a s

ide.

Wil

l it

fit

th

rou

gh

th

e d

oor

dia

gon

all

y?

Exp

lain

. N

o;

the g

reate

st

len

gth

th

at

will

fit

thro

ug

h t

he d

oo

r is

√ �

�

45 ≈

6.7

1 f

t.

11. S

CR

EE

N S

IZE

S T

he s

ize o

f a t

ele

vis

ion

is

measu

red

by t

he l

en

gth

of

the s

creen

’s

dia

gon

al.

a.

If a

tele

vis

ion

scr

een

measu

res

24 i

nch

es

hig

h a

nd

18 i

nch

es

wid

e,

wh

at

size

tele

vis

ion

is

it?

30-i

n.

tele

vis

ion

b.

Darl

a t

old

Tri

th

at

she h

as

a 3

5-i

nch

tele

vis

ion

. T

he h

eig

ht

of

the s

creen

is

21 i

nch

es.

Wh

at

is i

ts w

idth

? 28 i

n.

c.

Tri

told

Darl

a t

hat

he h

as

a 5

-in

ch h

an

dh

eld

tele

vis

ion

an

d t

hat

the s

creen

measu

res

2 i

nch

es

by 3

in

ches.

Is

this

a r

easo

nable

measu

re f

or

the s

creen

siz

e?

Exp

lain

. N

o;

if t

he s

cre

en

measu

res 2

in

. b

y 3

in

., t

hen

its

dia

go

nal

is o

nly

ab

ou

t 3.6

1 i

n.

12

4

b19

11

a

60

32

c

10-5

Answers (Lesson 10-5)

Co

pyrig

ht ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f Th

e M

cG

raw

-Hill C

om

pan

ies, In

c.

Chapter 10 A16 Glencoe Algebra 1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

34

Gle

ncoe A

lgeb

ra 1

Wo

rd

Pro

ble

m P

racti

ce

Pyth

ag

ore

an

Th

eo

rem

1. B

AS

EB

ALL

A

base

ball

dia

mon

d i

s a

squ

are

. E

ach

base

path

is

90 f

eet

lon

g.

Aft

er

a p

itch

, th

e c

atc

her

qu

ick

ly t

hro

ws

the b

all

fro

m h

om

e p

late

to a

team

mate

st

an

din

g b

y s

eco

nd

base

. F

ind

th

e

dis

tan

ce t

he b

all

tra

vels

. R

ou

nd

you

r an

swer

to t

he n

eare

st t

en

th.

127.3

ft

2. T

RIA

NG

LE

S E

ach

stu

den

t in

Mrs

. K

ell

y’s

geom

etr

y c

lass

con

stru

cted

a

un

iqu

e r

igh

t tr

ian

gle

fro

m d

rin

kin

g

stra

ws.

Mrs

. K

ell

y m

ad

e a

ch

art

wit

h t

he

dim

en

sion

s of

each

tri

an

gle

. H

ow

ever,

M

rs.

Kell

y m

ad

e a

mis

tak

e w

hen

re

cord

ing t

heir

resu

lts.

Wh

ich

resu

lt w

as

reco

rded

in

corr

ect

ly?

3. M

AP

S F

ind

th

e d

ista

nce

betw

een

Maco

n

an

d B

err

yvil

le.

Rou

nd

you

r an

swer

to t

he

neare

st t

en

th.

76.2

mi

4. T

ELE

VIS

ION

T

ele

vis

ion

s are

id

en

tifi

ed

by t

he d

iagon

al

measu

rem

en

t of

the

vie

win

g s

creen

. F

or

exam

ple

, a 2

7-i

nch

te

levis

ion

has

a d

iagon

al

scre

en

m

easu

rem

en

t of

27 i

nch

es.

Com

ple

te t

he c

hart

to f

ind

th

e s

creen

h

eig

ht

of

each

tele

vis

ion

giv

en

its

siz

e

an

d s

creen

wid

th.

Rou

nd

you

r an

swers

to

the n

eare

st w

hole

nu

mber.

So

urc

e:

Best

Buy

5. M

AN

UFA

CTU

RIN

G K

arl

work

s fo

r a

com

pan

y t

hat

man

ufa

ctu

res

car

part

s.

His

job i

s to

dri

ll a

hole

in

sp

heri

cal

steel

ball

s. T

he b

all

s an

d t

he h

ole

s h

ave t

he

dim

en

sion

s sh

ow

n o

n t

he d

iagra

m.

a.

How

deep

is

the h

ole

? 12 c

m

b.

Wh

at

wou

ld b

e t

he r

ad

ius

of

a b

all

w

ith

a s

imil

ar

hole

7 c

en

tim

ete

rs w

ide

an

d 2

4 c

en

tim

ete

rs d

eep

? 12.5

cm

d

Seco

nd

Base

Ho

me P

late

90

ft

x c

m

13

cm

5 c

m

Sid

e L

en

gth

s

S

tud

en

t a

b

c

Stu

den

t a

b

c

A

my

3

4

5

Fra

n

8

14

16

B

elin

da

7

24

25

Gus

5

12

13

E

mory

9

12

15

27

in

.

1 02345678

12

34

56

78

910

tens of miles

ten

s o

f m

iles

Cart

er

Cit

y

Maco

nH

am

ilto

n

Berr

yville

10-5

TV

siz

ew

idth

(in

.)h

eig

ht

(in

.)

19-inch

15

12

25-inch

21

14

32-inch

25

20

50-inch

40

30

Fra

n’s

Lesson 10-5

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

35

Gle

ncoe A

lgeb

ra 1

En

ric

hm

en

t

Pyth

ag

ore

an

Tri

ple

sR

eca

ll t

he P

yth

agore

an

Th

eore

m:

In a

rig

ht

tria

ngle

, th

e s

qu

are

of

the l

en

gth

of

the h

yp

ote

nu

se

is e

qu

al

to t

he s

um

of

the s

qu

are

s of

the l

en

gth

s of

the l

egs.

a

2 +

b2 =

c2

N

ote

th

at

c i

s th

e l

en

gth

of

the h

yp

ote

nu

se.

Th

e i

nte

gers

3,

4,

an

d 5

sati

sfy t

he

32 +

42 =

52

Pyth

agore

an

Th

eore

m a

nd

can

be t

he

9 +

16

= 2

5le

ngth

s of

the s

ides

of

a r

igh

t tr

ian

gle

. 25 =

25

Fu

rth

erm

ore

, fo

r an

y p

osi

tive i

nte

ger

n,

F

or

n =

2:

6

2 +

82 =

10

2

the n

um

bers

3n

, 4n

, an

d 5

n s

ati

sfy t

he

36 +

64 =

10

0P

yth

agore

an

Th

eore

m.

100 =

100

If t

hre

e n

um

bers

sati

sfy t

he P

yth

agore

an

Th

eore

m,

they a

re c

all

ed

a

Py

tha

go

rea

n t

rip

le.

Here

is

an

easy

way t

o f

ind

oth

er

Pyth

agore

an

tri

ple

s.

Th

e n

um

bers

a,

b,

an

d c

are

a P

yth

agore

an

tri

ple

if

a =

m2 -

n2,

b =

2m

n,

an

d c

= m

2 +

n2,

wh

ere

m a

nd

n a

re r

ela

tively

pri

me p

osi

tive i

nte

gers

an

d m

> n

.

C

ho

ose m

= 5

an

d n

= 2

.

a =

m2 -

n2

b =

2m

n

c =

m2 +

n2

Ch

eck

20

2 +

21

2

29

2

=

52 -

22

= 2

(5)(

2)

= 5

2 +

22

400 +

441

84

1

=

25

- 4

=

20

= 2

5 +

4

841 =

841

=

21

=

29

Exerc

ises

Use t

he f

oll

ow

ing

va

lues o

f m

an

d n

to

fin

d P

yth

ag

orea

n t

rip

les.

1. m

= 3

an

d n

= 2

2. m

= 4

an

d n

= 1

3. m

= 5

an

d n

= 3

5

, 12,

13

8,

15,

17

16,

30,

34

4. m

= 6

an

d n

= 5

5. m

= 1

0 a

nd

n =

7

6. m

= 8

an

d n

= 5

1

1,

60,

61

51,

140,

149

39,

80,

89

ACB a

c b

10-5

Exam

ple

Answers (Lesson 10-5)

Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he M

cG

raw

-Hill C

om

pan

ies,

Inc.

Chapter 10 A17 Glencoe Algebra 1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

36

Gle

ncoe A

lgeb

ra 1

Sp

read

sheet

Act

ivit

y

Pyth

ag

ore

an

Tri

ple

sA

Py

tha

go

rea

n t

rip

le i

s a s

et

of

thre

e w

hole

nu

mbers

th

at

sati

sfie

s th

e

equ

ati

on a

2 +

b2 =

c2, w

her

e c

is t

he

gre

ate

st n

um

ber

. Y

ou

can

use

a s

pre

adsh

eet

to i

nvest

igate

th

e p

att

ern

s in

Pyth

agore

an

tri

ple

s. A

prim

itiv

e P

yth

ag

orea

n

trip

le i

s a P

yth

agore

an

tri

ple

in

wh

ich

th

e n

um

bers

have n

o c

om

mon

fact

ors

oth

er

than

1.

A f

am

ily

of

Py

tha

go

rea

n t

rip

les i

s a p

rim

itiv

e P

yth

agore

an

trip

le a

nd

its

wh

ole

nu

mber

mu

ltip

les.

Th

e s

prea

dsh

eet

at

the r

igh

t p

ro

du

ces a

fa

mil

y o

f P

yth

ag

orea

n t

rip

les.

Ste

p 1

E

nte

r a P

yth

agore

an

tri

ple

in

to c

ell

s A

1,

A2,

an

d A

3.

Ste

p 2

U

se r

ow

s 2 t

hro

ugh

10 t

o f

ind

9

ad

dit

ion

al

Pyth

agore

an

tri

ple

s th

at

are

mu

ltip

les

of

the p

rim

itiv

e t

rip

le.

Form

at

the r

ow

s so

th

at

row

2 m

ult

ipli

es

the

nu

mbers

in

row

1 b

y 2

, ro

w 3

mu

ltip

lies

the n

um

bers

in

row

1 b

y 3

, an

d s

o o

n.

Exerc

ises

Use t

he s

prea

dsh

eet

of

fam

ilie

s o

f P

yth

ag

orea

n t

rip

les.

1. C

hoose

on

e o

f th

e t

rip

les

oth

er

than

(3,

4,

5)

from

th

e s

pre

ad

sheet.

Veri

fy

that

it i

s a P

yth

agore

an

tri

ple

. S

am

ple

an

sw

er:

Fo

r (6

, 8,

10),

62 +

82 =

36 +

64 o

r 100 =

10

2.

2. T

wo p

oly

gon

s are

sim

ila

r i

f th

ey a

re t

he s

am

e s

hap

e,

bu

t n

ot

nece

ssari

ly

the s

am

e s

ize.

For

tria

ngle

s, i

f tw

o t

rian

gle

s h

ave a

ngle

s w

ith

th

e s

am

e

measu

res

then

th

ey a

re s

imil

ar.

Use

a c

en

tim

ete

r ru

ler

to d

raw

tri

an

gle

s w

ith

measu

res

from

th

e s

pre

ad

sheet.

Do t

he t

rian

gle

s ap

pear

to b

e s

imil

ar?

See s

tud

en

ts’

wo

rk.;

yes

Ea

ch

of

the f

oll

ow

ing

is a

prim

itiv

e P

yth

ag

orea

n t

rip

le.

Use t

he

sp

rea

dsh

eet

to f

ind

tw

o P

yth

ag

orea

n t

rip

les i

n t

heir

fa

mil

ies.

3

. (5

, 12,

13)

Sam

ple

an

sw

er:

(10,

24,

26),

(15,

36,

39)

4.

(9,

40,

41)

Sam

ple

an

sw

er:

(18,

80,

82),

(27,

120,

123)

5

. (2

0,

21,

29)

Sam

ple

an

sw

er:

(40,

42,

58),

(60,

63,

87)

Tri

ple

s.x

ls

A

1 3 4 5 62 8 9 10

117

BC

3 6 9

12

15

18

21

24

27

30

4 8

12

16

20

24

28

32

36

40

5

10

15

20

25

30

35

40

45

50

The form

ula

in c

ell A

10 is

A1 *

10.

Sh

eet

1S

heet

2S

I

10-5

Lesson 10-6

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

37

Gle

ncoe A

lgeb

ra 1

F

ind

th

e d

ista

nce b

etw

een

th

e

po

ints

at

(-5,

2)

an

d (

4,

5).

d =

√ """""""""

(x2 -

x1)2

+ (y

2 -

y1)2

D

ista

nce F

orm

ula

=

√ """""""""

(4 -

(-

5))

2 +

(5 -

2)2

(x

1,

y1) =

(-

5,

2),

(x

2,

y2) =

(4,

5)

=

√ """

92 +

32

Sim

plif

y.

=

√ """

81 +

9

Evalu

ate

square

s a

nd s

implif

y.

=

√ ""

90

Th

e d

ista

nce

is

√ ""

90 ,

or

abou

t 9.4

9 u

nit

s.

Exam

ple

1

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Th

e D

ista

nce a

nd

Mid

po

int

Fo

rmu

las

Dis

tan

ce F

orm

ula

T

he P

yth

agore

an

Th

eore

m c

an

be u

sed

to d

eri

ve t

he D

ista

nce

Fo

rm

ula

sh

ow

n b

elo

w.

Th

e D

ista

nce

Form

ula

can

th

en

be u

sed

to f

ind

th

e d

ista

nce

betw

een

an

y t

wo p

oin

ts i

n t

he c

oord

inate

pla

ne.

Dis

tan

ce F

orm

ula

The d

ista

nce b

etw

een a

ny t

wo p

oin

ts w

ith c

oord

inate

s (

x1,

y1)

and (

x2,

y2)

is

giv

en b

y d

= √ """"""""

(x2 - x

1)2

+ (

y2 -

y1)2

J

ill

dra

ws a

lin

e

seg

men

t fr

om

po

int

(1,

4)

on

her

co

mp

ute

r s

creen

to

po

int

(98,

49).

H

ow

lo

ng

is t

he s

eg

men

t?

d =

√ """""""""

(x2 -

x1)2

+ (y

2 -

y1)2

=

√ """""""""

(98 -

1)2

+ (

49 -

4)2

=

√ """"

97

2 +

45

2

=

√ """"""

9409 +

2025

=

√ """

11,4

34

Th

e s

egm

en

t is

abou

t 106.9

3 u

nit

s lo

ng.

Exerc

ises

Fin

d t

he d

ista

nce b

etw

een

th

e p

oin

ts w

ith

th

e g

iven

co

ord

ina

tes.

1. (1

, 5),

(3,

1)

2. (0

, 0),

(6,

8)

3. (-

2, -

8),

(7, -

3)

2

√ "

5 ;

4.4

7

10

√ ""

106 ;

10.3

0

4. (6

, -

7),

(-

2,

8)

5. (1

, 5),

(-

8,

4)

6. (3

, -

4),

(-

4, -

4)

1

7

√ ""

82 ;

9.0

6

7

7. (-

1,

4),

(3,

2)

8. (0

, 0),

(-

3,

5)

9. (2

, -

6),

(-

7,

1)

2

√ "

5 ;

4.4

7

√ ""

34 ;

5.8

3

√ ""

130 ;

11.4

0

10. (-

2, -

5),

(0,

8)

11. (3

, 4),

(0,

0)

12. (3

, -

4),

(-

4, -

16)

√ ""

173 ;

13.1

5

5

√ ""

193 ;

13.8

9

Fin

d t

he p

ossib

le v

alu

es o

f a

if

the p

oin

ts w

ith

th

e g

iven

co

ord

ina

tes a

re t

he

ind

ica

ted

dis

tan

ce a

pa

rt.

13. (1

, a

), (

3, -

2);

d =

√ "

5

14. (0

, 0),

(a

, 4);

d =

5

15. (2

, -

1),

(a

, 3);

d =

5

-

1 o

r -

3

3 o

r -

3

-1 o

r 5

16. (1

, -

3),

(a

, 21);

d =

25

17. (1

, a

), (-

2,

4);

d =

3

18. (3

, -

4),

(-

4, a

); d

= √ ""

65

-

6 o

r 8

4

-8 o

r 0

10-6

Exam

ple

2

Answers (Lesson 10-5 and Lesson 10-6)

Co

pyrig

ht ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f Th

e M

cG

raw

-Hill C

om

pan

ies, In

c.

Chapter 10 A18 Glencoe Algebra 1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

38

Gle

ncoe A

lgeb

ra 1

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n(continued)

Th

e D

ista

nce a

nd

Mid

po

int

Fo

rmu

las

Mid

po

int

Fo

rmu

la

Th

e p

oin

t th

at

is e

qu

idis

tan

ce f

rom

both

of

the e

nd

poin

ts i

s ca

lled

th

e m

idp

oin

t. Y

ou

can

fin

d t

he c

oord

inate

s of

the m

idp

oin

t by u

sin

g t

he M

idp

oin

t F

orm

ula

.

Mid

po

int

Fo

rmu

la

The m

idpoin

t M

of

a lin

e s

egm

ent

with e

ndpoin

ts a

t (x

1,

y1)

and (

x2,

y2)

is

giv

en b

y M

( x1 +

x2

−

2

, y

1 +

y2

−

2

) .

F

ind

th

e c

oo

rd

ina

tes o

f th

e m

idp

oin

t o

f th

e s

eg

men

t w

ith

en

dp

oin

ts

at

(–2,

5)

an

d (

4,

9).

M ( x

1 +

x2 −

2

, y

1 +

y2 −

2

) M

idpoin

t F

orm

ula

=

M ( -

2 +

4 −

2

, 5

+ 9 −

2

) (x

1,

y1) =

(-

2,

5)

and (

x2,

y2) =

(4,

9)

=

M ( 2

−

2 ,

14 −

2 )

Sim

plif

y t

he n

um

era

tors

.

=

M (

1,

3)

Sim

plif

y.

Exerc

ises

Fin

d t

he c

oo

rd

ina

tes o

f th

e m

idp

oin

t o

f th

e s

eg

men

t w

ith

th

e g

iven

en

dp

oin

ts.

1. (1

, 6),

(3,

10)

2. (4

, -

2),

(0,

6)

3. (7

, 2),

(13, -

4)

(

2,

8)

(2,

2)

(10, -

1)

4. (-

1,

2),

(1,

0)

5. (-

3, -

3),

(5, -

11)

6. (0

, 8),

(-

6,

0)

(

1,

0)

(1, -

7)

(-

3,

4)

7. (4

, -

3),

(-

2,

3)

8. (9

, -

1),

(3, -

7)

9. (2

, -

1),

(8,

7)

(

1,

0)

(6, -

4)

(5,

3)

10. (1

, 4),

(-

3,

12)

11. (4

, 0),

(-

2,

6)

12. (1

, 9),

(7,

1)

(-

1,

8)

(1,

3)

(4,

5)

13. (1

2,

0),

(2, -

6)

14. (1

, 1),

(9, -

9)

15. (4

, 5),

(-

2, -

1)

(

7, -

3)

(5, -

4)

(1,

2)

16. (1

, -

14),

(-

5,

0)

17. (2

, 2),

(6,

8)

18. (-

7,

3),

(5, -

3)

(-

2, -

7)

(4,

5)

(-

1,

0)

10-6

Exam

ple

1

Lesson 10-6

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

39

Gle

ncoe A

lgeb

ra 1

Sk

ills

Pra

ctic

e

Th

e D

ista

nce a

nd

Mid

po

int

Fo

rmu

las

Fin

d t

he d

ista

nce b

etw

een

th

e p

oin

ts w

ith

th

e g

iven

co

ord

ina

tes.

1. (9

, 7),

(1,

1)

10

2

. (5

, 2),

(8, -

2)

5

3. (1

, -

3),

(1,

4)

7

4. (7

, 2),

(-

5,

7)

13

5. (-

6,

3),

(10,

3)

16

6. (3

, 3),

(-

2,

3)

5

7. (-

1, -

4),

(-

6,

0)

√ ""

41 ≈

6.4

0

8. (-

2,

4),

(5,

8)

√ ""

65 ≈

8.0

6

Fin

d t

he p

ossib

le v

alu

es o

f a

if

the p

oin

ts w

ith

th

e g

iven

co

ord

ina

tes a

re t

he

ind

ica

ted

dis

tan

ce a

pa

rt.

9. (-

2, -

5),

(a

, 7);

d =

13

a =

-7 o

r 3

10. (8

, -

2),

(5, a

); d

= 3

a

= -

2

11. (4

, a

), (

1,

6);

d =

5 a =

2 o

r 10

12. (a

, 3),

(5, -

1);

d =

5 a

= 2

or

8

13. (1

, 1),

(a

, 1);

d =

4 a =

-3 o

r 5

14. (2

, a

), (

2,

3);

d =

10

a =

-7 o

r 13

15. (a

, 2),

(-

3,

3);

d =

√ #

2 a

= -

4 o

r -

2

16. (-

5,

3),

(-

3, a

); d

= √ #

5 a

= 2

or

4

Fin

d t

he c

oo

rd

ina

tes o

f th

e m

idp

oin

t o

f th

e s

eg

men

t w

ith

th

e g

iven

en

dp

oin

ts.

17. (-

3,

4),

(-

2,

8)

(-2.5

, 6)

18. (5

, -

6),

(7, -

9)

(6, -

7.5

)

19. (4

, 2),

(8,

6)

(6,

4)

20. (5

, 2),

(3,

10)

(4,

6)

21. (1

2, -

1),

(4, -

11)

(8, -

6)

22. (-

3, -

1),

(-

11,

3)

(-7,

1)

23. (9

, 3),

(6, -

6)

(7.5

, -

1.5

) 24. (0

, -

4),

(8,

4)

(4,

0)

10-6

Answers (Lesson 10-6)

Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he M

cG

raw

-Hill C

om

pan

ies,

Inc.

Chapter 10 A19 Glencoe Algebra 1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

40

Gle

ncoe A

lgeb

ra 1

Practi

ce

Th

e D

ista

nce a

nd

Mid

po

int

Fo

rmu

las

Fin

d t

he d

ista

nce b

etw

een

th

e p

oin

ts w

ith

th

e g

iven

co

ord

ina

tes.

1. (4

, 7),

(1,

3)

5

2. (0

, 9),

(-

7,

-2)

√ �

�

170 ≈

13.0

4

3. (6

, 2),

+ (4

, 1

−

2 )

5

−

2 o

r 2.5

0

4. (-

1,

7),

+ ( 1

−

3 , 6

) 5

−

3 ≈

1.6

7

5.

( √ #

3 ,

3) ,

( 2 √

#

3 ,

5)

√ �

7 ≈

2.6

5

6.

( 2 √

#

2 ,

-1

) , ( 3

√ #

2 ,

3)

3 √

�

2 ≈

4.2

4

Fin

d t

he p

ossib

le v

alu

es o

f a

if

the p

oin

ts w

ith

th

e g

iven

co

ord

ina

tes a

re t

he

ind

ica

ted

dis

tan

ce a

pa

rt.

7. (4

, -

1),

(a

, 5);

d =

10

a =

-4 o

r 12

8

. (2

, -

5),

(a

, 7);

d =

15

a =

-7 o

r 11

9. (6

, -

7),

(a

, -

4);

d =

√ #

#

18 a

= 3

or

9

10. (-

4,

1),

(a

, 8);

d =

√ #

#

50 a

= -

5 o

r -

3

11. (8

, -

5),

(a

, 4);

d =

√ #

#

85 a

= 6

or

10

12. (-

9,

7),

(a

, 5);

d =

√ #

#

29 a

= -

14 o

r -

4

Fin

d t

he c

oo

rd

ina

tes o

f th

e m

idp

oin

t o

f th

e s

eg

men

t w

ith

th

e g

iven

en

dp

oin

ts.

13. (4

, -

6),

(3,

-9)

(3.5

, -

7.5

) 14. (-

3,

-8),

(-

7,

2)

(-5,

-3)

15. (0

, -

4),

(3,

2)

(1.5

, -

1)

16. (-

13,

-9),

(-

1,

-5)

(-7,

-7)

17.

(2, -

1

−

2 ) ,

(1, 1

−

2 ) (

1 1

−

2 ,

0)

18. ( 2

−

3 ,

-1

) , (2,

1

−

3 ) (1

1

−

3 ,

- 1

−

3 )

19. B

ASEB

ALL T

hre

e p

layers

are

warm

ing u

p f

or

a b

ase

ball

gam

e.

Pla

yer

B s

tan

ds

9 f

eet

to t

he r

igh

t an

d 1

8 f

eet

in f

ron

t of

Pla

yer

A.

Pla

yer

C s

tan

ds

8 f

eet

to t

he l

eft

an

d 1

3 f

eet

in

fron

t of

Pla

yer

A.

a

. D

raw

a m

od

el

of

the s

itu

ati

on

on

th

e c

oord

inate

gri

d.

Ass

um

e t

hat

Pla

yer

A i

s lo

cate

d a

t (0

, 0).

b

. T

o t

he n

eare

st t

en

th,

wh

at

is t

he d

ista

nce

betw

een

Pla

yers

A

an

d B

an

d b

etw

een

Pla

yers

A a

nd

C?

c.

Wh

at

is t

he d

ista

nce

betw

een

Pla

yers

B a

nd

C?

20. M

APS

M

ari

a a

nd

Jack

son

liv

e i

n a

dja

cen

t n

eig

hborh

ood

s. I

f th

ey s

up

eri

mp

ose

a

coord

inate

gri

d o

n t

he m

ap

of

their

neig

hborh

ood

s, M

ari

a l

ives

at

(-9,

1)

an

d J

ack

son

li

ves

at

(5,

-4).

x

y

O4

8

16

12 8 4

-8

-4

10-6

ab

ou

t 1.9

6 m

i

20.1

ft;

15.3

ft

17.7

ft

Lesson 10-6

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

41

Gle

ncoe A

lgeb

ra 1

Wo

rd

Pro

ble

m P

racti

ce

Th

e D

ista

nce a

nd

Mid

po

int

Fo

rmu

las

1. C

HESS

M

arg

are

t’s

last

tw

o r

em

ain

ing

chess

pie

ces

are

loca

ted

at

the c

en

ters

of

the s

qu

are

s at

op

posi

te c

orn

ers

of

the

board

. If

th

e c

hess

board

is

a s

qu

are

wit

h

8-i

nch

sid

es,

abou

t h

ow

far

ap

art

are

th

e

pie

ces?

Rou

nd

you

r an

swer

to t

he

neare

st t

en

th.

2. EN

GIN

EER

ING

T

od

d h

as

dra

wn

a

cul-

de-s

ac

for

a r

esi

den

tial

develo

pm

en

t p

lan

. H

e u

sed

a c

om

pass

to d

raw

th

e

cul-

de-s

ac

so t

hat

it w

ou

ld b

e c

ircu

lar.

O

n h

is b

luep

rin

t, t

he c

en

ter

of

the c

ul-

de-s

ac

has

coord

inate

s (-

1,

-1)

an

d a

p

oin

t on

th

e c

ircl

e i

s (2

, 3).

Wh

at

is t

he

rad

ius

of

the c

ul-

de-s

ac?

3. LA

ND

SC

APIN

G R

an

dy p

lott

ed

a

tria

ngu

lar

pati

o o

n a

lan

dsc

ap

e p

lan

for

a c

lien

t. W

hat

is t

he l

en

gth

of

fen

cin

g h

e

wil

l n

eed

alo

ng t

he p

ati

o e

dge t

hat

bord

ers

th

e p

rop

ert

y l

ine?

Rou

nd

you

r

an

swer

to t

he n

eare

st t

en

th.

4. U

TIL

ITIE

S T

he e

lect

ric

com

pan

y i

s ru

nn

ing s

om

e w

ires

acr

oss

an

op

en

fie

ld.

Th

e w

ire c

on

nect

s a u

tili

ty p

ole

at

(2,

14)

an

d a

seco

nd

uti

lity

pole

at

(7,

-8).

If

the

ele

ctri

c co

mp

an

y w

ish

es

to p

lace

a t

hir

d

pole

at

the m

idp

oin

t of

the t

wo p

ole

s, a

t w

hat

coord

inate

s sh

ou

ld t

he p

ole

be

pla

ced

?

5. M

AR

CH

ING

BA

ND

T

he O

hio

Sta

te

Un

ivers

ity m

arc

hin

g b

an

d p

erf

orm

s a

fam

ou

s on

-fie

ld s

pell

ing o

f O

-H-I

-O c

all

ed

“S

crip

t O

hio

”. S

om

eti

mes

they m

ust

ad

just

th

e u

sual

dim

en

sion

s of

the w

ord

to

fit

it

into

th

e l

imit

ed

gu

est

ban

d

perf

orm

an

ce a

rea.

Th

e d

iagra

m b

elo

w

show

s p

art

of

the a

dju

sted

dri

ll c

hart

. E

ach

poin

t re

pre

sen

ts o

ne b

an

d m

em

ber,

an

d t

he c

oord

inate

s are

in

yard

s.

a.

How

far

is t

he d

rum

majo

r fr

om

th

e

tuba p

layer

wh

o d

ots

th

e “

i”?

b.

Caro

l is

th

e b

an

d m

em

ber

at

the t

op

le

ft o

f th

e f

irst

O i

n O

hio

. S

he i

s lo

cate

d a

t (0

, 26).

How

far

aw

ay i

s C

aro

l fr

om

th

e t

uba p

layer?

Rou

nd

you

r an

swer

to t

he n

eare

st t

en

th.

dru

m m

ajo

rtu

ba p

layer

(20

, 1

3)

(32

, 1

0)

123456789

10

23

45

67

89

meters

meters

shru

bs

property line

house

treep

ati

o

( 2,

4)

( 6,

8)

( 5,

1)

10-6

9.9

in

.

5 u

nit

s

7.1

m

(4.5

, 3)

12.4

yd

23.9

yd

Answers (Lesson 10-6)

Co

pyrig

ht ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f Th

e M

cG

raw

-Hill C

om

pan

ies, In

c.

Chapter 10 A20 Glencoe Algebra 1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

42

Gle

ncoe A

lgeb

ra 1

En

rich

men

t

A S

pace-S

avin

g M

eth

od

Tw

o a

rran

gem

en

ts f

or

cook

ies

on

a 3

2 c

m b

y 4

0 c

m c

ook

ie

sheet

are

sh

ow

n a

t th

e r

igh

t. T

he c

ook

ies

have 8

-cm

dia

mete

rs

aft

er

they a

re b

ak

ed

. T

he c

en

ters

of

the c

ook

ies

are

on

th

e

vert

ices

of

squ

are

s in

th

e t

op

arr

an

gem

en

t. I

n t

he o

ther,

th

e

cen

ters

are

on

th

e v

ert

ices

of

equ

ilate

ral

tria

ngle

s. W

hic

h

arr

an

gem

en

t is

more

eco

nom

ical?

Th

e t

rian

gle

arr

an

gem

en

t is

more

eco

nom

ical,

beca

use

it

con

tain

s on

e m

ore

cook

ie.

In t

he s

qu

are

arr

an

gem

en

t, r

ow

s are

pla

ced

every

8 c

m.

At

wh

at

inte

rvals

are

row

s p

lace

d i

n t

he t

rian

gle

arr

an

gem

en

t?

Look

at

the r

igh

t tr

ian

gle

labele

d a

, b,

an

d c

. A

leg a

of

the

tr

ian

gle

is

the r

ad

ius

of

a c

ook

ie,

or

4 c

m.

Th

e h

yp

ote

nu

se c

is

the s

um

of

two r

ad

ii,

or

8 c

m.

Use

th

e P

yth

agore

an

th

eore

m t

o

fin

d b

, th

e i

nte

rval

of

the r

ow

s.

c

2 =

a2 +

b2

8

2 =

42 +

b2

64 -

16 =

b2

√ ""

48 =

b

4 √ "

3 =

b

b =

4 √ "

3 ≈

6.9

3

Th

e r

ow

s are

pla

ced

ap

pro

xim

ate

ly e

very

6.9

3 c

m.

So

lve e

ach

pro

ble

m.

1. S

up

pose

cook

ies

wit

h 1

0-c

m d

iam

ete

rs a

re a

rran

ged

in

th

e

tria

ngu

lar

patt

ern

sh

ow

n a

bove.

Wh

at

is t

he i

nte

rval

b o

f th

e r

ow

s? 8.6

6 c

m

2. F

ind

th

e d

iam

ete

r of

a c

ook

ie i

f th

e r

ow

s are

pla

ced

in

th

e

tria

ngu

lar

patt

ern

every

3 √ "

3 c

m.

6 c

m

3. D

esc

ribe o

ther

pra

ctic

al

ap

pli

cati

on

s in

wh

ich

th

is k

ind

of

tria

ngu

lar

patt

ern

can

be u

sed

to e

con

om

ize o

n s

pace

.

Sam

ple

an

sw

er:

packag

ing

can

s

{b=

?

21

co

okie

s

ca

{8

cm

20

co

okie

s

10-6

Lesson 10-7

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

43

Gle

ncoe A

lgeb

ra 1

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Sim

ilar

Tri

an

gle

s

Sim

ilar

Tri

an

gle

s %

RS

T i

s sim

ila

r t

o %

XY

Z.

Th

e a

ngle

s of

the

tw

o t

rian

gle

s h

ave e

qu

al

measu

re.

Th

ey a

re c

all

ed

co

rresp

on

din

g

an

gle

s.

Th

e s

ides

op

posi

te t

he c

orr

esp

on

din

g a

ngle

s are

call

ed

co

rresp

on

din

g s

ides.

30°

30°

60°

60°

S

ZX Y

RT

D

ete

rm

ine w

heth

er

the p

air

of

tria

ng

les i

s s

imil

ar.

Ju

sti

fy y

ou

r a

nsw

er.

Sin

ce c

orr

esp

on

din

g a

ngle

s d

o n

ot

have

the e

qu

al

measu

res,

th

e t

rian

gle

s are

n

ot

sim

ilar.

50°

75°

45°

89°

R

S

T

X

Y

Z

D

ete

rm

ine w

heth

er t

he p

air

of

tria

ng

les i

s s

imil

ar.

Ju

sti

fy y

ou

r

an

sw

er.

Th

e m

easu

re o

f ∠

G =

180° -

(90° +

45°)

= 4

5°.

T

he m

easu

re o

f ∠

I =

180° -

(45° +

45°)

= 9

0°.

S

ince

corr

esp

on

din

g a

ngle

s h

ave e

qu

al

measu

res,

%E

FG

∼ %

HIJ

.

90°

45°

45°

45°

F

G

H

J

I

E

Exerc

ises

Dete

rm

ine w

heth

er e

ach

pa

ir o

f tr

ian

gle

s i

s s

imil

ar.

Ju

sti

fy y

ou

r a

nsw

er.

1.

2.

3.

Y

es;

co

rresp

on

din

g

No

; co

rresp

on

din

g

Yes;

co

rresp

on

din

g

an

gle

s h

ave e

qu

al

a

ng

les d

o n

ot

have

an

gle

s h

ave e

qu

al

measu

res.

eq

ual

measu

res.

measu

res.

4.

5.

6.

Y

es;

co

rresp

on

din

g

Yes;

co

rresp

on

din

g

No

; co

rresp

on

din

g

an

gle

s h

ave e

qu

al

a

ng

les h

ave e

qu

al

an

gle

s d

o n

ot

have

measu

res.

measu

res.

eq

ual

measu

res.

20°

30°

12

0° 1

15°

45°

45°

80°

55°

40°

30°

30°

11

0°

90°

45°

60°

60°

30°

30°

30°

12

0°

10-7

Sim

ilar

Tri

an

gle

s

If t

wo t

riangle

s a

re s

imila

r, t

hen t

he

measure

s o

f th

eir c

orr

espondin

g s

ides

are

pro

port

ional and t

he m

easure

s o

f

their c

orr

espondin

g a

ngle

s a

re e

qual.

!

AB

C ∼

!D

EF

AB

−

DE =

BC

−

EF

= A

C

−

DF

A

BD

E

F

C

Exam

ple

1Exam

ple

2

Answers (Lesson 10-6 and Lesson 10-7)

Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he M

cG

raw

-Hill C

om

pan

ies,

Inc.

Chapter 10 A21 Glencoe Algebra 1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

44

Gle

ncoe A

lgeb

ra 1

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n(c

on

tin

ued

)

Sim

ilar

Tri

an

gle

s

Fin

d U

nkn

ow

n M

easu

res

If s

om

e o

f th

e m

easu

rem

en

ts a

re k

now

n,

pro

port

ion

s ca

n

be u

sed

to f

ind

th

e m

easu

res

of

the o

ther

sid

es

of

sim

ilar

tria

ngle

s.

IND

IREC

T M

EA

SU

REM

EN

T

�A

BC

∼ �

AED

in

th

e f

igu

re a

t th

e r

igh

t.

Fin

d t

he h

eig

ht

of

the a

pa

rtm

en

t b

uil

din

g.

Let

BC

= x

.

ED −

BC

= A

D −

AC

=

25 −

300

ED

= 7

, A

D =

25,

AC

= 3

00

25

x =

2100

Fin

d t

he c

ross p

roducts

.

x =

84

Th

e a

part

men

t bu

ild

ing i

s 84 m

ete

rs h

igh

.

Exerc

ises

Fin

d t

he m

issin

g m

ea

su

res f

or t

he p

air

of

sim

ila

r

tria

ng

les i

f �ABC

∼ �

DEF

.

1. c =

15,

d =

8,

e =

6,

f =

10 a

= 1

2;

b =

9

2. c =

20,

a =

12,

b =

8,

f =

15

d

= 9

; e

= 6

3. a

= 8

, d

= 8

, e =

6,

f =

7 b

= 6

; c

= 7

4. a

= 2

0,

d =

10,

e =

8,

f =

10

b

= 1

6;

c =

20

5. c =

5,

d =

10,

e =

8,

f =

8 a

= 2

5 −

4 ;

b =

5

6. a

= 2

5,

b =

20,

c =

15,

f =

12 d

= 2

0;

e =

16

7. b

= 8

, d

= 8

, e =

4,

f =

10

a

= 1

6;

c =

20

8. IN

DIR

EC

T M

EA

SU

RE

ME

NT

B

ruce

lik

es

to a

mu

se

his

bro

ther

by s

hin

ing a

fla

shli

gh

t on

his

han

d a

nd

mak

ing a

sh

ad

ow

on

th

e w

all

. H

ow

far

is i

t fr

om

the f

lash

ligh

t to

th

e w

all

? 51.6

in

. o

r 4.3

ft

9. IN

DIR

EC

T M

EA

SU

RE

ME

NT

A

fore

st r

an

ger

use

s si

mil

ar

tria

ngle

s to

fin

d t

he h

eig

ht

of

a t

ree.

Fin

d t

he h

eig

ht

of

the t

ree.

60 f

t

12

ft

20

ft

x

10

0 f

t

5 in

.6 in.

x in.

4 f

t

Note: N

ot

dra

wn t

o s

cale

A

ca

be

fd

CD

F

EB

25

m

Note: N

ot

dra

wn t

o s

cale

27

5 m

7 m

ADE

CB

x

10-7

Exam

ple

7 −

x

Lesson 10-7

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

45

Gle

ncoe A

lgeb

ra 1

Sk

ills

Pra

ctic

e

Sim

ilar

Tri

an

gle

s

PR

Q

q

rp

SU

T

t

us

52°

52°

63°

65°

FE

H

J

K

G4

0°

40°

45°

F

EG

H

J

K

60°

60°

57°

60°

XV

W

U

Z

Y

40°

50°

A

DF

E C

B

10-7

Dete

rm

ine w

heth

er e

ach

pa

ir o

f tr

ian

gle

s i

s s

imil

ar.

Ju

sti

fy y

ou

r a

nsw

er.

1.

2.

Y

es; ∠

A =

∠D

= 9

0°;

∠B

=

No

; ∠

Y =

180° -

(60° +

60°)

= 6

0°.

180° -

(90° +

40°)

= 5

0° =

∠E

;

Sin

ce �

UV

W h

as a

57°

an

gle

, b

ut

∠F =

180° -

(90° +

50°)

= 4

0° =

�

XY

Z d

oes n

ot,

co

rresp

on

din

g

∠C

. S

ince t

he c

orr

esp

on

din

g

an

gle

s d

o n

ot

all h

ave e

qu

al

an

gle

s h

ave e

qu

al

measu

res,

m

easu

res,

an

d t

he t

rian

gle

s a

re n

ot

�A

BC

∼ �

DE

F.

sim

ilar.

3.

4.

N

o; ∠

F =

180° -

(45° +

40°)

=

Yes; ∠

G =

180° -

(65° +

52°)

= 6

3° =

95°.

Sin

ce �

HJK

has a

90°

∠

K; ∠

J =

180° -

(63° +

52°)

= 6

5° =

an

gle

, b

ut �

EFG

do

es n

ot,

∠

F; ∠

E =

∠H

= 5

2°.

Sin

ce t

he

co

rresp

on

din

g a

ng

les d

o n

ot

all

co

rresp

on

din

g a

ng

les h

ave e

qu

al

have e

qu

al

measu

res,

an

d t

he

m

easu

res, �

EFG

∼ �

HJK

.

tria

ng

les a

re n

ot

sim

ilar.

Fin

d t

he m

issin

g m

ea

su

res f

or t

he p

air

of

sim

ila

r

tria

ng

les i

f �PQR

∼ �

STU

.

5.

r =

4,

s =

6,

t =

3,

u =

2 p

= 1

2,

q =

6

6.

t =

8,

p =

21,

q =

14,

r =

7 u

= 4

, s =

12

7.

p =

15,

q =

10,

r =

5,

s =

6 t =

4,

u =

2

8.

p =

48,

s =

16,

t =

8,

u =

4 r =

12,

q =

24

9.

q =

6,

s =

2,

t =

3 −

2 ,

u =

1 −

2 r =

2,

p =

8

10.

p =

3,

q =

2,

r =

1,

u =

1 −

3 s =

1,

t =

2 −

3

11.

p =

14,

q =

7,

u =

2.5

, t =

5 r =

3.5

, s =

10

12.

r =

6,

s =

3,

t =

21 −

8 ,

u =

9 −

4 p

= 8

, q

= 7

Answers (Lesson 10-7)

Co

pyrig

ht ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f Th

e M

cG

raw

-Hill C

om

pan

ies, In

c.

Chapter 10 A22 Glencoe Algebra 1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

46

Gle

ncoe A

lgeb

ra 1

Practi

ce

Sim

ilar

Tri

an

gle

sD

ete

rm

ine w

heth

er e

ach

pa

ir o

f tr

ian

gle

s i

s s

imil

ar.

Ju

sti

fy y

ou

r a

nsw

er.

1.

2.

Y

es; ∠

Q =

∠T =

90°;

∠P

=

No

; ∠

C =

180° -

(47° +

80°)

= 5

3°.

180° -

(90° +

31°)

= 5

9° =

∠S

;

Sin

ce !

FG

H h

as a

56°

an

gle

, b

ut

∠U

= 1

80° -

(90° +

59°)

= 3

1° =

!

CD

E d

oes n

ot,

co

rresp

on

din

g

∠R

. S

ince t

he c

orr

esp

on

din

g

an

gle

s d

o n

ot

all h

ave e

qu

al

an

gle

s h

ave e

qu

al

measu

res,

m

easu

res,

an

d t

he t

rian

gle

s a

re n

ot

!P

QR

, !

STU

. sim

ilar.

Fin

d t

he m

issin

g m

ea

su

res f

or t

he p

air

of

sim

ila

r

tria

ng

les i

f !ABC

∼ !

DEF

.

3. c =

4, d

= 1

2, e =

16, f =

8 a

= 6

, b

= 8

4. e =

20, a

= 2

4, b

= 3

0, c =

15 d

= 1

6,

f =

10

5. a

= 1

0, b =

12, c =

6, d

= 4

e =

4.8

, f =

2.4

6. a

= 4

, d

= 6

, e =

4, f =

3 c =

2,

b =

8 −

3

7. b

= 1

5, d

= 1

6, e =

20, f =

10

a =

12,

c =

15 −

2

8. a

= 1

6, b =

22, c =

12, f =

8 d

= 3

2 −

3 ,

e =

44 −

3

9. a

= 5

−

2 , b =

3, f =

11 −

2 ,

e =

7 c

= 3

3 −

14 ,

d =

35 −

6

10. c =

4, d

= 6

, e =

5.6

25, f =

12 a

= 2

, b

= 1

.875

11. S

HA

DO

WS

S

up

pose

you

are

sta

nd

ing n

ear

a b

uil

din

g a

nd

you

wan

t to

kn

ow

its

heig

ht.

Th

e b

uil

din

g c

ast

s a 6

6-f

oot

shad

ow

. Y

ou

cast

a 3

-foot

shad

ow

. If

you

are

5 f

eet

6 i

nch

es

tall

, h

ow

tall

is

the b

uil

din

g?

12. M

OD

ELS

T

russ

bri

dges

use

tri

an

gle

s in

th

eir

su

pp

ort

beam

s. M

oll

y m

ad

e a

mod

el

of

a

tru

ss b

rid

ge i

n t

he s

cale

of

1 i

nch

= 8

feet.

If

the h

eig

ht

of

the t

rian

gle

s on

th

e m

od

el

is

4.5

in

ches,

wh

at

is t

he h

eig

ht

of

the t

rian

gle

s on

th

e a

ctu

al

bri

dge?

DF

E

e

fd

AC

B

b

ca

80°

47°

47°

56°

EH

F

GD

C

31°

59°

RQ

STU

P

10-7

121 f

t

36 f

t

Lesson 10-7

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

47

Gle

ncoe A

lgeb

ra 1

Wo

rd

Pro

ble

m P

racti

ce

Sim

ilar

Tri

an

gle

s

1. C

RA

FTS

L

ayla

is

wan

ts t

o b

uy a

set

of

sim

ilar

magn

ets

for

her

refr

igera

tor

door.

Layla

fin

ds

the m

agn

ets

belo

w f

or

sale

at

a l

oca

l sh

op

. W

hic

h t

wo a

re s

imil

ar?

B a

nd

C

2. EX

HIB

ITIO

NS

T

he w

orl

d’s

larg

est

can

dle

was

dis

pla

yed

at

the 1

897 S

tock

holm

Exh

ibit

ion

. S

up

pose

Lars

measu

red

th

e

len

gth

of

the s

had

ow

it

cast

at

11:0

0 A

.M.

an

d f

ou

nd

th

at

it w

as

12 f

eet.

Su

pp

ose

that

imm

ed

iate

ly a

fter

this

, h

e m

easu

red

to f

ind

th

at

a n

earb

y 2

5-f

oot

ten

t p

ole

cast

a s

had

ow

5 f

eet

lon

g.

How

tall

was

the w

orl

d’s

larg

est

can

dle

?

3. LA

ND

MA

RK

S T

he T

oy a

nd

Min

iatu

re

Mu

seu

m o

f K

an

sas

Cit

y d

isp

lays

a

min

iatu

re r

ep

lica

of

Georg

e W

ash

ingto

n’s

Mou

nt

Vern

on

man

sion

. T

he m

inia

ture

hou

se i

s 10 f

eet

lon

g,

6 f

eet

wid

e,

8 f

eet

tall

, an

d h

as

22 r

oom

s. T

he s

cale

of

the

mod

el

to t

he o

rigin

al

is o

ne i

nch

to o

ne

foot.

If

the r

oof

gable

of

the m

inia

ture

has

dim

en

sion

s as

show

n o

n t

he d

iagra

m

belo

w,

wh

at

is t

he h

eig

ht

of

the r

oof

gable

on

th

e o

rigin

al

Mou

nt

Vern

on

man

sion

?

So

urce:

Mount

Vern

on

4. S

UR

VE

YIN

G S

urv

eyors

use

pro

pert

ies

of

tria

ngle

s in

clu

din

g s

imil

ari

ty a

nd

th

e

Pyth

agore

an

Th

eore

m t

o f

ind

un

kn

ow

n

dis

tan

ces.

Use

th

e d

imen

sion

s on

th

e

dia

gra

m t

o f

ind

th

e u

nk

now

n d

ista

nce

x

acr

oss

th

e l

ak

e.

80 m

5. PU

ZZLES

T

he f

igu

re b

elo

w s

how

s an

an

cien

t C

hin

ese

movable

pu

zzle

call

ed

a

tan

gra

m.

It h

as

7 p

iece

s th

at

can

be

reco

nfi

gu

red

to p

rod

uce

an

en

dle

ss

nu

mber

of

desi

gn

s an

d p

ictu

res.

Ass

um

e t

hat

the s

ide l

en

gth

of

this

tan

gra

m s

qu

are

is

√$ 2 c

m.

Leave y

ou

r

an

swers

as

sim

pli

fied

rad

ical

exp

ress

ion

s.

a

. Wh

at

are

th

e s

ide l

en

gth

s of

tria

ngle

s

1 a

nd

2?

b

. Wh

at

are

th

e s

ide l

en

gth

s of

tria

ngle

7?

√ &

2 − 2

cm

, √ &

2 − 2

cm

, 1 c

m

c.

Wh

at

are

th

e s

ide l

en

gth

s of

tria

ngle

s

3 a

nd

5?

AB

C

7cm

6cm

5cm

3cm

4cm

8cm

3 f

t

1.9

ft

1.9

ft

hei

gh

t

NP

M

x

Q

80 m

40 m

120 m

0

lake

1

2

3

4

5

67

10-7

1 −

2 c

m,

1 −

2 c

m,

√ &

2 −

2 c

m

60 f

t

14 f

t1 c

m,

1 c

m,

√ &

2 c

m

Answers (Lesson 10-7)

Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he M

cG

raw

-Hill C

om

pan

ies,

Inc.

Chapter 10 A23 Glencoe Algebra 1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

48

Gle

ncoe A

lgeb

ra 1

En

rich

men

t

A C

uri

ou

s C

on

str

ucti

on

Man

y m

ath

em

ati

cian

s h

ave b

een

in

tere

sted

in

ways

to

con

stru

ct t

he n

um

ber π

. H

ere

is

on

e s

uch

geom

etr

ic c

on

stru

ctio

n.

In t

he d

raw

ing,

tria

ngle

s A

BC

an

d A

DE

are

rig

ht

tria

ngle

s. T

he l

en

gth

of

−−−

AD

equ

als

th

e l

en

gth

of

−−

AC

an

d −−

FB

is

para

llel

to −−−

EG

.

Th

e l

en

gth

of

−−−

BG

giv

es

a d

eci

mal

ap

pro

xim

ati

on

of

the f

ract

ion

al

part

of π t

o s

ix d

eci

mal

pla

ces.

Fo

llo

w t

he s

tep

s t

o f

ind

th

e l

en

gth

of

−−

BG

. R

ou

nd

to s

ev

en

decim

al

pla

ces.

1. U

se t

he l

en

gth

of

−−−

BC

an

d t

he P

yth

agore

an

Th

eore

m t

o f

ind

th

e l

en

gth

of

−−

AC

.

A

C =

√ ""

""

1 2 +

( 7

−

8 ) 2

= 1

.3287682

2. F

ind

th

e l

en

gth

of

−−−

AD

.

A

D =

AC

= 1

.3287682

3. U

se t

he l

en

gth

of

−−−

AD

an

d t

he P

yth

agore

an

Th

eore

m t

o f

ind

th

e l

en

gth

of

−−

AE

.

A

E =

√ ""

""

""

(A

D ) 2

+ ( 1

−

2 ) 2

= 1

.4197271

4. T

he s

ides

of

the s

imil

ar

tria

ngle

s F

ED

an

d D

EA

are

in

pro

port

ion

. S

o,

FE −

0.5

= 0

.5 −

AE

.

Fin

d t

he l

en

gth

of

−−

FE

.

F

E =

1

−

4(A

E)

= 0

.1760902

5. F

ind

th

e l

en

gth

of

−−

AF

.

A

F =

AE

- F

E =

1.2

436369

6. T

he s

ides

of

the s

imil

ar

tria

ngle

s A

FB

an

d A

EG

are

in

pro

port

ion

. S

o,

AF −

AE

= A

B −

AG

.

Fin

d t

he l

en

gth

of

−−−

AG

.

A

G =

AB

· A

E

−

AF

= A

E

−

AF =

1.1

415929

7. N

ow

, fi

nd

th

e l

en

gth

of

−−−

BG

.

B

G =

AG

- A

B =

AG

- 1

= 0

.1415929

8. T

he v

alu

e o

f π

to s

even

deci

mal

pla

ces

is 3

.1415927.

Com

pare

th

e f

ract

ion

al

part

of

pi

wit

h t

he l

en

gth

of

−−−

BG

.

0

.1415929 -

0.1

415927 =

0.0

000002,

an

err

or

of

less t

han

1 p

art

in

a m

illio

n

7 – 8

1 – 2

C

E DG

BA

F

1

10-7

Lesson 10-8

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

49

Gle

ncoe A

lgeb

ra 1

10-8

Tri

go

no

metr

ic R

ati

os

Tri

gon

om

etr

y i

s th

e s

tud

y o

f re

lati

on

ship

s of

the a

ngle

s an

d t

he

sid

es

of

a r

igh

t tr

ian

gle

. T

he t

hre

e m

ost

com

mon

tri

gon

om

etr

ic r

ati

os

are

th

e s

ine

, co

sin

e,

an

d t

an

gen

t.

sin

e o

f ∠

A = le

g o

pposite ∠

A

−

hypote

nuse

sin

e o

f ∠

B = le

g o

pposite ∠

B

−

hypote

nuse

sin

A = a

−

c

sin

B = b

−

c

cosin

e o

f ∠

A = le

g a

dja

cent to

∠A

−−

hypote

nuse

cosin

e o

f ∠

B = le

g a

dja

cent to

∠B

−−

hypote

nuse

cos A

= b

−

c

cos B

= a

−

c

tangent

of ∠

A =

leg o

pposite ∠

A

−−

le

g a

dja

cent to

∠A

tangent

of ∠

B =

leg o

pposite ∠

B

−−

le

g a

dja

cent to

∠B

tan A

= a

−

b

tan B

= b

−

a

F

ind

th

e v

alu

es o

f th

e t

hree t

rig

on

om

etr

ic r

ati

os f

or a

ng

le A

.

Ste

p 1

U

se t

he P

yth

agore

an

Th

eore

m t

o f

ind

BC

.

a2 +

b2 = c

2

Pyth

agore

an T

heore

m

a2 +

82 = 1

02

b =

8 a

nd c

= 1

0

a

2 +

64 = 1

00

Sim

plif

y.

a2 = 3

6

Subtr

act

64 f

rom

both

sid

es.

a = 6

T

ake t

he s

quare

root

of

each s

ide.

Ste

p 2

U

se t

he s

ide l

en

gth

s to

wri

te t

he t

rigon

om

etr

ic r

ati

os.

sin

A = o

pp −

hyp =

6 −

10 =

3 −

5

co

s A

=

ad

j −

hyp =

8 −

10 =

4 −

5

tan

A = o

pp −

ad

j = 6

−

8 = 3

−

4

Exerc

ises

Fin

d t

he v

alu

es o

f th

e t

hree t

rig

on

om

etr

ic r

ati

os f

or a

ng

le A

.

1.

8

17

2.

3

5

3.

24

7

sin

A =

15 −

17 ,

co

s A

=

8 −

17 ,

sin

A =

7 −

25 ,

co

s A

= 2

4 −

25 ,

tan

A =

15 −

8

ta

n A

=

7 −

24

Use a

ca

lcu

lato

r t

o f

ind

th

e v

alu

e o

f ea

ch

trig

on

om

etr

ic r

ati

o t

o t

he n

ea

rest

ten

-th

ou

sa

nd

th.

4. s

in 4

0°

5. co

s 25°

0.9

063

6. ta

n 8

5°

11.4

301

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Tri

go

no

metr

ic R

ati

os

Exam

ple

a

10 8

0.6

428

b

a

c

sin

A =

4

−

5 ,

co

s A

= 3

−

5 ,

tan

A =

4

−

3

Answers (Lesson 10-7 and Lesson 10-8)

Co

pyrig

ht ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f Th

e M

cG

raw

-Hill C

om

pan

ies, In

c.

Chapter 10 A24 Glencoe Algebra 1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

50

Gle

ncoe A

lgeb

ra 1

10-8

Use

Tri

go

no

metr

ic R

ati

os

Wh

en

you

fin

d a

ll o

f th

e u

nk

now

n m

easu

res

of

the s

ides

an

d a

ngle

s of

a r

igh

t tr

ian

gle

, you

are

so

lvin

g t

he t

ria

ng

le.

You

can

fin

d t

he m

issi

ng

measu

res

of

a r

igh

t tr

ian

gle

if

you

kn

ow

th

e m

easu

re o

f tw

o s

ides

of

the t

rian

gle

, or

the

measu

re o

f on

e s

ide a

nd

th

e m

easu

re o

f on

e a

cute

an

gle

.

So

lve t

he t

ria

ng

le.

Ro

un

d e

ach

sid

e l

en

gth

to

th

e n

ea

rest

ten

th.

Ste

p 1

F

ind

th

e m

easu

re o

f ∠

B.

Th

e s

um

of

the m

easu

res

of

the a

ngle

s in

a t

rian

gle

is

180.

180° −

(90° +

38°)

= 5

2°

Th

e m

easu

re o

f ∠

B i

s 52°.

Ste

p 2

F

ind

th

e m

easu

re o

f −−

AB

. B

eca

use

you

are

giv

en

th

e

measu

re o

f th

e s

ide a

dja

cen

t to

∠ A

an

d a

re f

ind

ing

the m

easu

re o

f th

e h

yp

ote

nu

se,

use

th

e c

osi

ne r

ati

o.

co

s 38° =

13 −

c

D

efinitio

n o

f cosin

e

c c

os

38° =

13

M

ultip

ly e

ach s

ide b

y c

.

c =

13 −

cos

38°

Div

ide e

ach s

ide b

y s

in 4

1°.

So t

he m

easu

re o

f −−

AB

is

abou

t 16.5

.

Ste

p 3

F

ind

th

e m

easu

re o

f −−−

BC

. B

eca

use

you

are

giv

en

th

e m

easu

re o

f th

e

sid

e a

dja

cen

t to

∠ A

an

d a

re f

ind

ing t

he m

easu

re o

f th

e s

ide o

pp

osi

te

∠ A

, u

se t

he t

an

gen

t ra

tio.

tan

38° =

a −

13

Definitio

n o

f ta

ngent

13 t

an

38° =

a

Multip

ly e

ach s

ide b

y 1

3.

10.2

≈ a

U

se a

calc

ula

tor.

So t

he m

easu

re o

f −−−

BC

is

abou

t 10.2

.

Exerc

ises

So

lve e

ach

rig

ht

tria

ng

le.

Ro

un

d e

ach

sid

e l

en

gth

to

th

e n

ea

rest

ten

th.

1.

9

ab 30°

2.

b

8c

44°

3.

16c

b

56°

∠B

= 6

0°,

AC

≈ 7

.8,

∠A

= 6

0°,

AC

≈ 7

.7,

∠B

= 3

4°,

AC

≈ 1

9.3

,B

C =

4.5

A

B ≈

11.1

A

B ≈

10.8

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n (c

on

tin

ued

)

Tri

go

no

metr

ic R

ati

os

Exam

ple

13

ac

38°

Lesson 10-8

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

51

Gle

ncoe A

lgeb

ra 1

Fin

d t

he v

alu

es o

f th

e t

hree t

rig

on

om

etr

ic r

ati

os f

or a

ng

le A

.

1.

77

85

2.

15

9

sin

A =

7

7

−

85 , c

os A

= 36

−

85 , t

an

A =

77

−

36

sin

A =

4

−

5 , c

os A

= 3

−

5 , t

an

A =

4

−

3

3.

10

24

4.

15

8

sin

A =

12

−

13 , c

os A

=

5

−

13 , t

an

A =

12

−

5

sin

A =

8

−

17 , c

os A

= 1

5

−

17 , t

an

A =

8

−

15

Use a

ca

lcu

lato

r t

o f

ind

th

e v

alu

e o

f ea

ch

trig

on

om

etr

ic r

ati

o t

o t

he

nea

rest

ten

-th

ou

sa

nd

th.

5. s

in 1

8°

0.3

090

6.

cos

68°

0.3

746

7. t

an

27°

0.5

095

8. c

os

60°

0.5

9.

tan

75°

3.7

321

10. s

in 9

° 0.1

564

So

lve e

ach

rig

ht

tria

ng

le.

Ro

un

d e

ach

sid

e l

en

gth

to

th

e n

ea

rest

ten

th.

11.

13

17°

12.

6

55°

∠A

= 7

3°,

AB

= 1

3.6

, A

C =

4.0

∠

B =

35°,

AB

= 1

0.5

, B

C =

8.6

Fin

d m

∠J

fo

r e

ach

rig

ht

tria

ng

le t

o t

he n

ea

rest

deg

ree.

13.

6

5

14.

19

11

40°

55°

10-8

Sk

ills

Pra

ctic

e

Tri

go

no

metr

ic R

ati

os

Answers (Lesson 10-8)

Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he M

cG

raw

-Hill C

om

pan

ies,

Inc.

Chapter 10 A25 Glencoe Algebra 1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

52

Gle

ncoe A

lgeb

ra 1

Fin

d t

he v

alu

es o

f th

e t

hree t

rig

on

om

etr

ic r

ati

os f

or a

ng

le A

.

1.

7297

2.

36

15

sin

A =

65

−

97 ,

co

s A

= 7

2

−

97 ,

tan

A =

65

−

72

sin

A =

36

−

39 ,

co

s A

= 1

5

−

39 ,

tan

A =

36

−

15

Use a

ca

lcu

lato

r t

o f

ind

th

e v

alu

e o

f ea

ch

trig

on

om

etr

ic r

ati

o t

o t

he n

ea

rest

ten

-th

ou

sa

nd

th.

3. t

an

26°

0.4

877

4. s

in 5

3°

0.7

986

5. c

os

81°

0.1

564

So

lve e

ach

rig

ht

tria

ng

le.

Ro

un

d e

ach

sid

e l

en

gth

to

th

e n

ea

rest

ten

th.

6.

67°

22

7.

9

29°

∠B

= 2

3°,

AB

= 2

3.9

, A

C =

9.3

∠

A =

61°,

AB

= 1

0.3

, B

C =

5.0

Fin

d m

∠J

fo

r e

ach

rig

ht

tria

ng

le t

o t

he n

ea

rest

deg

ree.

8.

11

5

9.

12

18

24°

42°

10. SU

RV

EY

ING

If

poin

t A

is

54 f

eet

from

th

e

tree,

an

d t

he a

ngle

betw

een

th

e g

rou

nd

at

poin

t A

an

d t

he t

op

of

the t

ree i

s 25°,

fin

d

the h

eig

ht h

of

the t

ree.

25.2

ft

10-8

Practi

ce

Tri

go

no

metr

ic R

ati

os

25°

54

ft

h

Lesson 10-8

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

10

53

Gle

ncoe A

lgeb

ra 1

10-8

Wo

rd

Pro

ble

m P

racti

ce

Tri

go

no

metr

ic R

ati

os

1. W

ASH

ING

TO

N M

ON

UM

EN

TJean

nie

is t

ryin

g t

o d

ete

rmin

e t

he h

eig

ht

of

the

Wash

ingto

n M

on

um

en

t. I

f p

oin

t A

is

765 f

eet

from

th

e m

on

um

en

t, a

nd

th

e

an

gle

betw

een

th

e g

rou

nd

an

d t

he t

op

of

the m

on

um

en

t at

poin

t A

is

36°,

fin

d t

he

heig

ht h

of

the m

on

um

en

t to

th

e

neare

st f

oot.

765 ft3

6°

h

2. A

IRPLA

NES

A

pil

ot

tak

es

off

fro

m a

run

way a

t an

an

gle

of

20º

an

d m

ain

tain

s

that

an

gle

un

til

it i

s at

its

cru

isin

g

alt

itu

de o

f 2500 f

eet.

Wh

at

hori

zon

tal

dis

tan

ce h

as

the p

lan

e t

ravele

d w

hen

it

reach

es

its

cru

isin

g a

ltit

ud

e?

3. TR

UC

K R

AM

PS

A

movin

g c

om

pan

y u

ses

an

11-f

oot-

lon

g r

am

p t

o u

nlo

ad

fu

rnit

ure

from

a t

ruck

. If

th

e b

ed

of

the t

ruck

is

3

feet

above t

he g

rou

nd

, w

hat

is t

he a

ngle

of

incl

ine o

f th

e r

am

p t

o t

he n

eare

st

degre

e?

4. SPEC

IAL T

RIA

NG

LES

W

hil

e

invest

igati

ng r

igh

t tr

ian

gle

KLM

,

Merc

ed

es

fin

ds

that

cos M

= s

in M

.

Wh

at

is t

he m

easu

re o

f an

gle

M?

5. TELEV

ISIO

NS

Tele

vis

ion

s are

com

mon

ly

sized

by m

easu

rin

g t

heir

dia

gon

al.

A

com

mon

siz

e f

or

wid

esc

reen

pla

sma T

Vs

is 4

2 i

nch

es.

42

’’h

h1

6 9

a

. A

wid

esc

reen

tele

vis

ion

has

a 1

6:9

asp

ect

rati

o,

that

is,

the s

creen

wid

th

is 1

6 −

9

tim

es

the s

creen

heig

ht.

Use

th

e

Pyth

agore

an

Th

eore

m t

o w

rite

an

equ

ati

on

an

d s

olv

e f

or

the h

eig

ht h

of

the t

ele

vis

ion

in

in

ches.

b

. Use

th

e i

nfo

rmati

on

fro

m p

art

a t

o

solv

e t

he r

igh

t tr

ian

gle

.

c.

Wh

at

wou

ld t

he m

easu

re o

f an

gle

A b

e

on

a s

tan

dard

tele

vis

ion

wit

h a

4:3

asp

ect

rati

o?

( 16 −

9

h) 2

+ h

2 =

42

2;

h =

20.6

in

.

wid

th =

36.6

in

., ∠

A = 2

9°,

∠B

= 6

1°

16°

45°

6869 f

t

556 f

t

37°

Answers (Lesson 10-8)

ERROR: undefined

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