11-1 skills practice the polygons in the panels of the ... ( –2, 4) f(2, 4) x y o l(6, 0) m(1 ......

Skills PracticeArea of Parallelograms

NAME ______________________________________________ DATE ____________ PERIOD _____

11-111-1

© Glencoe/McGraw-Hill 613 Glencoe Geometry

Less

on

11-

1

Find the perimeter and area of each parallelogram. Round to the nearest tenth ifnecessary.

1. 2.

3. 4.

5. 6.

Find the area of each figure.

7. 8.

COORDINATE GEOMETRY Given the coordinates of the vertices of a quadrilateral,determine whether it is a square, a rectangle, or a parallelogram. Then find thearea of the quadrilateral.

9. A(�4, 2), B(�1, 2), C(�1, �1), 10. P(�3, 3), Q(1, 3), R(1, �3),D(�4, �1) S(�3, �3)

11. D(�5, 1), E(7, 1), F(4, �4), 12. R(2, 3), S(4, 10), T(12, 10),G(�8, �4) U(10, 3)

8

222

2 2

2

263

3

4

6

6

1

1

1

1122

22

1

18.5 km

9 km

3.4 m

26 in.

22 in.

45�

14 yd

7 yd45�

5.5 ft

4 ft

60�

20 cm

30 cm60�

Reading to Learn MathematicsAreas of Triangles, Trapezoids, and Rhombi

NAME ______________________________________________ DATE ____________ PERIOD _____

11-211-2


Less

on

11-

2

Pre-Activity How is the area of a triangle related to beach umbrellas?

Read the introduction to Lesson 11-2 at the top of page 601 in your textbook.

Classify the polygons in the panels of the beach umbrella.

Reading the Lesson1. Match each area formula from the first column with the corresponding polygon in the

second column.

a. A � �w i. triangle

b. A � �12�d1d2 ii. parallelogram

c. A � s2 iii. trapezoid

d. A � �12�h(b1 � b2) iv. rhombus

e. A � �12�bh v. square

f. A � bh vi. rectangle

2. Determine whether each statement is always, sometimes, or never true. In each case,explain your reasoning.

a. The area of a square is half the product of its diagonals.

b. The area of a triangle is half the product of two of its sides.

c. You can find the area of a rectangle by multiplying base times height.

d. You can find the area of a rectangle by multiplying the lengths of any two of its sides.

e. The area of a trapezoid is the product of its height and the sum of the bases.

f. The square of the length of a side of a square is equal to half the product of itsdiagonals.

Helping You Remember3. A good way to remember a new geometric formula is to state it in words. Write a short

sentence that tells how to find the area of a trapezoid in a way that is easy to remember.


Areas of Similar TrianglesYou have learned that if two triangles are similar, the ratio of the lengths of correspondingaltitudes is equal to the ratio of the lengths of a pair of corresponding sides. However, thereis a different relationship between the areas of the two triangles.

Theorem If two triangles are similar, the ratio of their areas is the square of the ratio ofthe lengths of a pair of corresponding sides.

Triangle II is k times larger than Triangle I. Thus, its base is k times as large as that of Triangle I and its height is k times as large as that ofTriangle I.

�ssiiddee

ooff�

�

III

� � �kbb� or �

k1�

�aarreeaa

ooff�

�

III

� � or �k1

2�

Solve.

1. �DEF � �GHJ, HJ � 16, and EF � 8. 2. In the figure below, P�Q� || B�C�. The area ofThe area of �GHJ is 40. Find the area �ABC is 72. Find the area of �APQ.of �DEF.

3. Two similar triangles have areas of 16 and 36. The length of a side of the smallertriangle is 10 feet. Find the length of the corresponding side of the larger triangle.

4. Find the ratio of the areas of two similar triangles if the lengths of two correspondingsides of the triangles are 3 centimeters and 5 centimeters.

B C

QP

8

4 3

6

A

H J

GE F

D

Triangle II

area �II � �12�(kb)(kh)

� �12�k2bh

Triangle I

area �I � �12�bh�

12�k2bh��

�12�bh

kh

kb

h

b

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

11-211-2

Skills PracticeAreas of Regular Polygons and Circles

NAME ______________________________________________ DATE ____________ PERIOD _____

11-311-3


Less

on

11-

3

Find the area of each regular polygon. Round to the nearest tenth.

1. a pentagon with a perimeter of 45 feet

2. a hexagon with a side length of 4 inches

3. a nonagon with a side length of 8 meters

4. a triangle with a perimeter of 54 centimeters

Find the area of each circle. Round to the nearest tenth.

5. a circle with a radius of 6 yards

6. a circle with a diameter of 18 millimeters

Find the area of each shaded region. Assume that all polygons are regular. Roundto the nearest tenth.

7. 8.

9. 10.

5 cm4 ft

8 m

4 m

3 in.

Skills PracticeAreas of Irregular Figures

NAME ______________________________________________ DATE ____________ PERIOD _____

11-411-4


Less

on

11-

4

Find the area of each figure. Round to the nearest tenth if necessary.

1. 2.

3. 4.

5. 6.

7. 8.

x

y

O

D(–2, –2)

G(4, 1)

H(2, –2)

E(–2, 4) F(2, 4)

x

y

O L(6, 0)

M(1, –3)

J(1, 4) K(5, 4)

x

y

O

Q(3, 5)

R(6, 7)P(0, 7)

T(3, 2)

U(0, 0) S(6, 0)x

y

O

D(8, 5)

E(8, 0)

C(3, 8)

B(3, 5)

A(0, 0)

30

15

8

8

7

3

5

12

20

Reading to Learn MathematicsAreas of Irregular Figures

NAME ______________________________________________ DATE ____________ PERIOD _____

11-411-4


Less

on

11-

4

Pre-Activity How do windsurfers use area?

Read the introduction to Lesson 11-4 at the top of page 617 in your textbook.

How do you think the areas of the figures outlined in the picture of the sailare related?

Reading the Lesson1. Use dashed segments to show how each figure can be subdivided into figures for which you

have learned area formulas. Name the smaller figures that you have formed as specificallyas possible and indicate whether any of them are congruent to each other.

a. b. c.

2. In the figure, B is the midpoint of ABC�. Complete the following steps to derive a formula for the area of the shaded region in terms of the radius r of the circle.

The area of circle P is .

m�ABC � because

.

mAB�� mBC� because

.

A�B� � B�C� because

.

Therefore, �ABC is a(n) triangle.

AC � , so AB � and BC � .

The area of �ABC is �12� � � � .

Therefore, the area of the shaded region is given by

A � � � .

Helping You Remember3. Rolando is having trouble remembering when to subtract an area when finding the area

of an irregular figure. How can you help him remember?

B C

A

P

Study Guide and InterventionGeometric Probability

NAME ______________________________________________ DATE ____________ PERIOD _____

11-511-5


Less

on

11-

5

Geometric Probability The probability that a point in a figure will lie in a particularpart of the figure can be calculated by dividing the area of the part of the figure by the area ofthe entire figure. The quotient is called the geometric probability for the part of the figure.

If a point in region A is chosen at random, then the probability P(B) that the point is in region B, which is in the interior of region A, is

P(B) � �aarreeaa

ooff

rreeggiioonn

BA

�.

Darts are thrown at a circular dartboard.If a dart hits the board, what is the probability that the dart lands in the bull’s-eye?Area of bull’s-eye: A � �(2)2 or 4�

Area of entire dartboard: A � �(10)2 or 100�

The probability of landing in the bull’s-eye is

� �140

�0��

� �215� or 0.04.

Find the probability that a point chosen at random lies in the shaded region.Round to the nearest hundredth if necessary.

1. 2.

3. 4.

5. 6. 2 cm3 cm1 cm

88

6

6

6

6

88

24

24

12

12

area of bull’s-eye��area of dartboard

2 in.

4 in.

4 in.

ExampleExample

ExercisesExercises


Sectors and Segments of Circles A sector of a circleis a region of a circle bounded by a central angle and itsintercepted arc. A segment of a circle is bounded by a chord and its arc. Geometric probability problems sometimes involvesectors or segments of circles.

If a sector of a circle has an area of A square units, a central angle

measuring N °, and a radius of r units, then A � �3N60��r 2.

A regular hexagon is inscribed in a circle with diameter 12. Findthe probability that a point chosen at random in the circle lies in the shadedregion.The area of the shaded segment is thearea of sector AOF � the area of �AOF.

Area of sector AOF � �3N60��r2

� �36600��(62)

� 6�

Area of �AOF � �12�bh

� �12�(6)(3�3�)

� 9�3�The shaded area is 6� � 9�3� or about 3.26.

The probability is �ar

aeraea

ofosfecgirmcleent

� � �33.62�6

� or about 0.03.

Find the probability that a point in the circle chosen at random lies in the shadedregion. Round to the nearest hundredth.

1. 2. 3.

4. 5. 6.

44.94

60�

56120�

4110�

120�

60�4 in.120�

120�70�

50�

10 cm

12A D

EF

O

B C

sector

segment

Study Guide and Intervention (continued)

Geometric Probability

NAME ______________________________________________ DATE ____________ PERIOD _____

11-511-5

ExercisesExercises

ExampleExample

© Glencoe/McGraw-Hill A3 Glencoe Geometry

An

swer

s

Skil

ls P

ract

ice

Are

a o

f P

aral

lelo

gra

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

11-1

11-1

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Lesson 11-1

Fin

d t

he

per

imet

er a

nd

are

a of

eac

h p

aral

lelo

gram

.Rou

nd

to

the

nea

rest

ten

th i

fn

eces

sary

.

1.2.

100

cm,5

19.6

cm

219

ft,

19.1

ft2

3.4.

42 y

d,6

9.3

yd2

96 in

.,40

4.5

in2

5.6.

13.6

m,1

1.6

m2

55 k

m,1

66.5

km

2

Fin

d t

he

area

of

each

fig

ure

.

7.8.

14 u

nit

s258

un

its2

CO

OR

DIN

ATE

GEO

MET

RYG

iven

th

e co

ord

inat

es o

f th

e ve

rtic

es o

f a

qu

adri

late

ral,

det

erm

ine

wh

eth

er i

t is

a s

qu

are

,a r

ecta

ngl

e,or

a p

ara

llel

ogra

m.T

hen

fin

d t

he

area

of

the

qu

adri

late

ral.

9.A

(�4,

2),B

(�1,

2),C

(�1,

�1)

,10

.P(�

3,3)

,Q(1

,3),

R(1

,�3)

,D

(�4,

�1)

S(�

3,�

3)

squ

are,

9 u

nit

s2re

ctan

gle

,24

un

its2

11.D

(�5,

1),E

(7,1

),F

(4,�

4),

12.R

(2,3

),S

(4,1

0),T

(12,

10),

G(�

8,�

4)U

(10,

3)

par

alle

log

ram

,60

un

its2

par

alle

log

ram

,56

un

its2

8 22

2222

26

33

4

66

1 1

1

11

2 22 2

1

18.5

km

9 km

3.4

m

26 in

.

22 in

.

45�

14 y

d

7 yd

45�

5.5

ft

4 ft 60

�

20 c

m

30 c

m60

�

©G

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Fin

d t

he

per

imet

er a

nd

are

a of

eac

h p

aral

lelo

gram

.Rou

nd

to

the

nea

rest

ten

th i

fn

eces

sary

.

1.2.

3.

32 m

,47.

6 m

236

cm

,56.

6 cm

234

.1 in

.,50

in2

Fin

d t

he

area

of

each

fig

ure

.

4.5.

44 u

nit

s265

un

its2

CO

OR

DIN

ATE

GEO

MET

RYG

iven

th

e co

ord

inat

es o

f th

e ve

rtic

es o

f a

qu

adri

late

ral,

det

erm

ine

wh

eth

er i

t is

a s

qu

are

,a r

ecta

ngl

e,or

a p

ara

llel

ogra

m.T

hen

fin

d t

he

area

of

the

qu

adri

late

ral.

6.C

(�4,

�1)

,D(�

4,2)

,F(1

,2),

G(1

,�1)

7.W

(2,2

),X

(1,�

2),Y

(�2,

�2)

,Z(�

1,2)

rect

ang

le,1

5 u

nit

s2p

aral

lelo

gra

m,1

2 u

nit

s2

8.M

(0,4

),N

(4,6

),O

(6,2

),P

(2,0

)9.

P(�

5,2)

,Q(4

,2),

R(5

,5),

S(�

4,5)

squ

are,

20 u

nit

s2p

aral

lelo

gra

m,2

7 u

nit

s2

FRA

MIN

GF

or E

xerc

ises

10–

12,u

se t

he

foll

owin

g in

form

atio

n.

A r

ecta

ngu

lar

post

er m

easu

res

42 i

nch

es b

y 26

in

ches

.A f

ram

e sh

op f

itte

d th

e po

ster

wit

h a

hal

f-in

ch m

at b

orde

r.

10.F

ind

the

area

of

the

post

er.

1092

in2

11.F

ind

the

area

of

the

mat

bor

der.

69 in

2

12.S

upp

ose

the

wal

l is

mar

ked

wh

ere

the

post

er w

ill

han

g.T

he

mar

ked

area

in

clu

des

anad

diti

onal

12-

inch

spa

ce a

rou

nd

the

post

er a

nd

fram

e.F

ind

the

tota

l w

all

area

th

at h

asbe

en m

arke

d fo

r th

e po

ster

.34

17 in

2

773

34

31

33

3 3

2

2

22

8

92

2

2

22

2

1

1

4 4

4

10 in

.

45�

10 c

m

8 cm 45

�

5 m

11 m

60�

Pra

ctic

e (

Ave

rag

e)

Are

a o

f P

aral

lelo

gra

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

11-1

11-1

Answers (Lesson 11-1)


An

swer

s

Readin

g t

o L

earn

Math

em

ati

csA

reas

of T

rian

gle

s,Tr

apez

oid

s,an

d R

ho

mb

i

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

11-2

11-2

©G

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Lesson 11-2

Pre-

Act

ivit

yH

ow i

s th

e ar

ea o

f a

tria

ngl

e re

late

d t

o b

each

um

bre

llas

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 11

-2 a

t th

e to

p of

pag

e 60

1 in

you

r te

xtbo

ok.

Cla

ssif

y th

e po

lygo

ns

in t

he

pan

els

of t

he

beac

h u

mbr

ella

.Is

osc

eles

tri

ang

les

and

iso

scel

es t

rap

ezo

ids

Rea

din

g t

he

Less

on

1.M

atch

eac

h a

rea

form

ula

fro

m t

he

firs

t co

lum

n w

ith

th

e co

rres

pon

din

g po

lygo

n i

n t

he

seco

nd

colu

mn

.

a.A

��w

vii.

tria

ngl

e

b.

A�

�1 2� d1d

2iv

ii.p

aral

lelo

gram

c.A

�s2

vii

i.tr

apez

oid

d.

A�

�1 2� h(b

1�

b 2)

iiiiv

.rh

ombu

s

e.A

��1 2� b

hi

v.sq

uar

e

f.A

�bh

iivi

.rec

tan

gle

2.D

eter

min

e w

het

her

eac

h s

tate

men

t is

alw

ays,

som

etim

es,o

r n

ever

tru

e.In

eac

h c

ase,

expl

ain

you

r re

ason

ing.

Fo

r ex

pla

nat

ion

s,sa

mp

le a

nsw

ers

are

giv

en.

a.T

he

area

of

a sq

uar

e is

hal

f th

e pr

odu

ct o

f it

s di

agon

als.

Alw

ays;

a sq

uar

e is

arh

om

bus,

so y

ou

can

use

th

e rh

om

bus

form

ula

.

b.

Th

e ar

ea o

f a

tria

ngl

e is

hal

f th

e pr

odu

ct o

f tw

o of

its

sid

es.

So

met

imes

;th

is is

tru

e o

nly

fo

r a

rig

ht

tria

ng

le.

c.Yo

u ca

n fi

nd t

he a

rea

of a

rec

tang

le b

y m

ulti

plyi

ng b

ase

tim

es h

eigh

t.A

lway

s;a

rect

angl

e is

a p

aral

lelo

gram

,so

you

can

use

the

para

llelo

gram

form

ula.

Ifth

e le

ngth

of

a re

ctan

gle

is u

sed

as t

he b

ase,

then

the

wid

th is

the

hei

ght.

d.

You

can

fin

d th

e ar

ea o

f a

rect

angl

e by

mu

ltip

lyin

g th

e le

ngt

hs

of a

ny

two

of i

ts s

ides

.S

om

etim

es;

this

is t

rue

on

ly f

or

a sq

uar

e.O

ther

wis

e,yo

u m

ust

use

tw

oco

nse

cuti

vesi

des

,no

t an

y tw

o s

ides

.e.

Th

e ar

ea o

f a

trap

ezoi

d is

th

e pr

odu

ct o

f it

s h

eigh

t an

d th

e su

m o

f th

e ba

ses.

Nev

er;

the

area

is o

ne-

hal

f th

e p

rod

uct

of

its

hei

gh

t an

d t

he

sum

of

the

bas

es.

f.T

he

squ

are

of t

he

len

gth

of

a si

de o

f a

squ

are

is e

qual

to

hal

f th

e pr

odu

ct o

f it

sdi

agon

als.

Alw

ays;

a sq

uar

e is

a r

ho

mbu

s,so

th

e fo

rmu

las

for

a sq

uar

ean

d a

rh

om

bus

mu

st g

ive

the

sam

e an

swer

wh

enev

er t

he

rho

mbu

s is

asq

uar

e.

Hel

pin

g Y

ou

Rem

emb

er3.

A g

ood

way

to

rem

embe

r a

new

geo

met

ric

form

ula

is

to s

tate

it

in w

ords

.Wri

te a

sh

ort

sen

ten

ce t

hat

tel

ls h

ow t

o fi

nd

the

area

of

a tr

apez

oid

in a

way

th

at i

s ea

sy t

o re

mem

ber.

Sam

ple

an

swer

:A

vera

ge

the

len

gth

s o

f th

e b

ases

an

d m

ult

iply

by

the

hei

gh

t.

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Are

as o

f S

imila

r Tri

ang

les

You

hav

e le

arn

ed t

hat

if

two

tria

ngl

es a

re s

imil

ar,t

he

rati

o of

th

e le

ngt

hs

of c

orre

spon

din

gal

titu

des

is e

qual

to

the

rati

o of

th

e le

ngt

hs

of a

pai

r of

cor

resp

ondi

ng

side

s.H

owev

er,t

her

eis

a d

iffe

ren

t re

lati

onsh

ip b

etw

een

th

e ar

eas

of t

he

two

tria

ngl

es.

Th

eore

mIf

tw

o tr

ian

gles

are

sim

ilar

,th

e ra

tio

of t

hei

r ar

eas

is t

he

squ

are

of t

he

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o of

the

len

gth

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of c

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e II

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mes

lar

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rian

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hu

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rian

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d it

s h

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t is

kti

mes

as

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e as

th

at o

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or �k 1�

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ve.

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,HJ

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elow

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|| B�C�

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e ar

ea o

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ind

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ave

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and

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he

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0 fe

et.F

ind

the

len

gth

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the

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ng

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larg

er t

rian

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t

4.F

ind

the

rati

o of

th

e ar

eas

of t

wo

sim

ilar

tri

angl

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f th

e le

ngt

hs

of t

wo

corr

espo

ndi

ng

side

s of

th

e tr

ian

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are

3 c

enti

met

ers

and

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met

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8

43

6

A

HJ

GE

F

D

Tri

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area

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kb)(

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area

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rich

men

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NA

ME

____

____

____

____

____

____

____

____

____

____

____

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AT

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____

____

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ER

IOD

____

_

11-2

11-2



An

swer

s

Skil

ls P

ract

ice

Are

as o

f R

egu

lar

Po

lyg

on

s an

d C

ircl

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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AT

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____

____

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ER

IOD

____

_

11-3

11-3

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Lesson 11-3

Fin

d t

he

area

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each

reg

ula

r p

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oun

d t

o th

e n

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st t

enth

.

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pen

tago

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ith

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erim

eter

of

45 f

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2.a

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sid

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inch

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non

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tria

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ith

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erim

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of

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area

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cle.

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nd

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the

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th.

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ith

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ion

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ll p

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lar.

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nd

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7.8.

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in2

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.

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8 m

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.

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r p

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.

1.a

non

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wit

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per

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7 m

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.7 m

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cle.

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ll p

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5.6.

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.7 in

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7.8.

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DIS

PLA

YS

For

Exe

rcis

es 9

an

d 1

0,u

se t

he

foll

owin

g in

form

atio

n.

A d

ispl

ay c

ase

in a

jew

elry

sto

re h

as a

bas

e in

th

e sh

ape

of a

reg

ula

r oc

tago

n.T

he

len

gth

of

each

sid

e of

th

e ba

se i

s 10

in

ches

.Th

e ow

ner

s of

th

e st

ore

plan

to

cove

r th

e ba

se i

n b

lack

velv

et.

9.F

ind

the

area

of

the

base

of

the

disp

lay

case

.

abo

ut

482.

8 in

2

10.F

ind

the

nu

mbe

r of

squ

are

yard

s of

fab

ric

nee

ded

to c

over

th

e ba

se.

abo

ut

0.37

yd

2

9 m

25 ft

4.4

in.

12 c

mPra

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e (

Ave

rag

e)

Are

as o

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lar

Po

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s an

d C

ircl

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

11-3

11-3



Skil

ls P

ract

ice

Are

as o

f Ir

reg

ula

r F

igu

res

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

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ER

IOD

____

_

11-4

11-4

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Lesson 11-4

Fin

d t

he

area

of

each

fig

ure

.Rou

nd

to

the

nea

rest

ten

th i

f n

eces

sary

.

1.2.

150

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14.0

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Fin

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ure

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LAN

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For

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,use

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ne

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he

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ical

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re s

how

s th

e di

men

sion

s of

th

e po

nd

and

the

wal

kway

.

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ind

the

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pon

d to

th

e n

eare

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.

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ind

the

area

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the

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13

13

20

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9

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22

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20

Pra

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Are

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reg

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r F

igu

res

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

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____

____

__P

ER

IOD

____

_

11-4

11-4



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csA

reas

of

Irre

gu

lar

Fig

ure

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

11-4

11-4

©G

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Lesson 11-4

Pre-

Act

ivit

yH

ow d

o w

ind

surf

ers

use

are

a?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 11

-4 a

t th

e to

p of

pag

e 61

7 in

you

r te

xtbo

ok.

How

do

you

th

ink

the

area

s of

th

e fi

gure

s ou

tlin

ed i

n t

he

pict

ure

of

the

sail

are

rela

ted?

Sam

ple

an

swer

:Th

e ar

eas

get

sm

alle

r as

yo

u m

ove

furt

her

up

th

e sa

il.T

he

area

of

the

tria

ng

le is

sm

alle

r th

an t

he

area

of

any

of

the

trap

ezo

ids.

Rea

din

g t

he

Less

on

1.U

se d

ashe

d se

gmen

ts t

o sh

ow h

ow e

ach

figu

re c

an b

e su

bdiv

ided

into

fig

ures

for

whi

ch y

ouha

ve le

arne

d ar

ea f

orm

ulas

.Nam

e th

e sm

alle

r fi

gure

s th

at y

ou h

ave

form

ed a

s sp

ecif

ical

lyas

pos

sibl

e an

d in

dica

te w

het

her

an

y of

th

em a

re c

ongr

uen

t to

eac

h o

ther

.S

amp

le a

nsw

ers

are

giv

en.

a.b

.c.

rect

ang

le a

nd

sq

uar

e an

d

rect

ang

le a

nd

is

osc

eles

tri

ang

letw

o c

on

gru

ent

two

co

ng

ruen

t is

osc

eles

tri

ang

les

sem

icir

cles

2.In

the

fig

ure,

Bis

the

mid

poin

t of

AB

C�

.Com

plet

e th

e fo

llow

ing

step

s to

der

ive

a fo

rmul

a fo

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f th

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aded

reg

ion

in t

erm

s of

the

rad

ius

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the

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Th

e ar

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Pis

.

m�

AB

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beca

use

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o A

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and

BC

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e ar

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.

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eref

ore,

the

area

of

the

shad

ed r

egio

n i

s gi

ven

by

A�

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.

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pin

g Y

ou

Rem

emb

er3.

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ando

is

hav

ing

trou

ble

rem

embe

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g w

hen

to

subt

ract

an

are

a w

hen

fin

din

g th

e ar

eaof

an

irr

egu

lar

figu

re.H

ow c

an y

ou h

elp

him

rem

embe

r?S

amp

le a

nsw

er:

Su

btr

act

wh

en t

her

e is

an

ind

enta

tio

n,o

r a

ho

le in

th

e fi

gu

re.

�� 2��

1 �r2

r2�1 2� �

r2

r2r�

2�r�

2�

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or

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� �2r2��

or

r�2�

2r

iso

scel

es r

igh

t o

r 45

°-45

°-90

°co

ng

ruen

t,th

eir

corr

esp

on

din

g c

ho

rds

are

con

gru

ent

Sam

ple

an

swer

:If

tw

o m

ino

r ar

cs o

f a

circ

le a

re

Sam

ple

an

swer

:B

is t

he

mid

po

int

of

AB

C�

(def

init

ion

of

mid

po

int)

Sam

ple

an

swer

:It

is a

n in

scri

bed

an

gle

th

at in

terc

epts

a s

emic

ircl

e90

�r2

BC

A

P

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Aer

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urv

eyo

rs a

nd

Are

aM

any

lan

d re

gion

s h

ave

irre

gula

r sh

apes

.Aer

ial

surv

eyor

s of

ten

u

se c

oord

inat

es w

hen

fin

din

g ar

eas

of s

uch

reg

ion

s.T

he

coor

din

ate

met

hod

des

crib

ed i

n t

he

step

s be

low

can

be

use

d to

fin

d th

e ar

ea

of a

ny

poly

gon

al r

egio

n.S

tudy

how

th

is m

eth

od i

s u

sed

to f

ind

the

area

of

the

regi

on a

t th

e ri

ght.

Ste

p 1

Lis

t th

e or

dere

d pa

irs

for

the

vert

ices

in

cou

nte

r-cl

ockw

ise

orde

r,re

peat

ing

the

firs

t or

dere

d pa

ir a

t th

e bo

ttom

of

the

list

.

Ste

p 2

Fin

d D

,th

e su

m o

f th

e do

wn

war

d di

agon

al p

rodu

cts

(fro

m

left

to

righ

t).

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(5�

5)�

(2�

1)�

(2�

3)�

(6�

7)�

25�

2�

6�

42 o

r 75

Ste

p 3

Fin

d U

,th

e su

m o

f th

e u

pwar

d di

agon

al p

rodu

cts

(fro

m l

eft

to r

igh

t).

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(2�

7)�

(2�

5)�

(6�

1)�

(5�

3)�

14�

10�

6�

15 o

r 45

Ste

p 4

Use

th

e fo

rmu

la A

��1 2� (

D�

U)

to f

ind

the

area

.

A�

�1 2� (D

�U

)

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75�

45)

��1 2� (

30)

or 1

5

Th

e ar

ea i

s 15

squ

are

un

its.

Cou

nt

the

nu

mbe

r of

squ

are

un

its

encl

osed

by

the

poly

gon

.Doe

s th

is r

esu

lt s

eem

rea

son

able

?

Use

th

e co

ord

inat

e m

eth

od t

o fi

nd

th

e ar

ea o

f ea

ch r

egio

n i

n s

qu

are

un

its.

1.2.

3.

20 u

nit

s214

un

its2

34 u

nit

s2

x

y

O

x

y

Ox

y

O

(5, 7

)

(2, 5

)

(2, 1

)

(6, 3

)

(5, 7

)

x

y

O

(2, 5

)

(2, 1

)

(6, 3

)

(5, 7

)

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

11-4

11-4



Stu

dy G

uid

e a

nd I

nte

rven

tion

Geo

met

ric

Pro

bab

ility

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

11-5

11-5

©G

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5G

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eom

etry

Lesson 11-5

Geo

met

ric

Pro

bab

ility

The

pro

babi

lity

tha

t a

poin

t in

a f

igur

e w

ill l

ie in

a p

arti

cula

rpa

rt o

f th

e fi

gure

can

be

calc

ulat

ed b

y di

vidi

ng t

he a

rea

of t

he p

art

of t

he f

igur

e by

the

are

a of

the

enti

re f

igur

e.T

he q

uoti

ent

is c

alle

d th

e ge

omet

ric

pro

bab

ilit

yfo

r th

e pa

rt o

f th

e fi

gure

.

If a

poin

t in

reg

ion

Ais

cho

sen

at r

ando

m,

then

the

pro

babi

lity

P(B

) th

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____

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11-5

11-5

Exer

cises

Exer

cises

Exam

ple

Exam

ple


11-1 skills practice the polygons in the panels of the ... ( –2, 4) f(2, 4) x y o l(6, 0) m(1 ......

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