\ \ Name: Mr, ~'..Mi (4V\S(.u..(,r ~ Date:-----

.i ß7 ..... _ DILATIONS _ ~~f\.ec,.\-t'o~'\ I.\ #' Jr- <:... / - ·Ífo-V\.S: \4--\?c:>V\ · - ~.-\-G...~ova

We've seen three fundamental transformations so far that are rigid motions, i.e. transformations that do not shinge--thH&~ nor the si-ze ofan olaject. l'hsrn 1s an important transfönnatrnn, tfioygfi, tAat ym,1 @ßcotmtsr often in the real world, the dilation. This transformation enlarges or shrinks an object by some factor, but leaves its overall shape the same.

Exercise #1: Given the following images, determine the dilation scale factor, k.

(a)

l. 1 •

• • :1 (b)

• ·1: :1 .- '/ ' -t- 1vl

31 '-I \\'\. i.. ~ - , ,5ìV"\ I 'l\e-k. -

o,75ì"' )( \<.. =- I · 5,"" { "'~ a-7

In order for phones, tablets, computers, and other electronic devices to be able to enlarge or shrink images, the process needs to be able to be described mathematically. The definition of a dilation involves two things, a center of dilation and a dilation scale factor. The technical definition is the following:

1-6i()Ck.S l .5 'f... h ;;:.. r

DILATIONS

A dilation D with a center at point Canda scale factor of k (where k must be positive) is a fonction that has as its input a point in the plane, A, and has as its output the image point, A', such that:

I. If A is the center of dilation, i.e. C, then D ( C) = C, in other words the dilation does not move the center. ¡ 2. If A is any other point and D (A)= A' then A' is a point on the ray CA such that CA'= k · CA .

Exercise #2: If Cis the center of the dilation, use your ruler to find the image of A after a dilation by a factor of (a) k = 2 and (b) k = 0.5. Label the first image point A' and the second A". Leave ,,, -6 all marks. Write down equations relating the length of CA to ~ • 'd- 1/

"' P< and CA" v

Length Equations:

Exercise #3: Given MEC shown below, we would like to dilate it using A as a center with a scale factor of 2.

/ca) Using a straightedge and a compass, find the image of B and C and label them as B' and C'. Leave all marks.

Connect B' and C' to form B'C'. I ~

(b) Why wouldn't A change its location under this d ·iati~

. 14--- ÌS +4-, C...b'\+d""

cF J:\o.A,oV" 8 A

(c) Explain why any points along either line AC gr ~ will still lie on th~e lines after the dilation?

ße.c~ bo¼ AL ~J 1îß Con¼"' ·-\-k c__-e_/4-e_r- e£- e\: \Q_~~

( d) D is the midpoint of BC. Find its image under this same dilation. Does it fall on B' C'? Does it fall at the

midpoint of B 'C'? Check using your ruler.

'/ß I .PÔ<-\\ 5 Or'\ j I

13 c.- ªPf<ò)L 7ö ().\rl'J. Ye,~ , is -fu.,~ :e\ Pº~ ·"' t.. aF- B 'c.,'

(e) How does the length of B'C' compare to the length of BC? Check using your ruler or compass. -- ß'e..' í.:s (¼ \e¾.'\ ~ Y.. ~ ß'é.' \}

10"1-'- =- ¡L-{O / ìzë: ( f) What else seems to be true about the segments BC and B 'C'? How can you verify this experimentally?

'(Y) eAS "re of 0 .. Y\ j \~<s (l.,'('€.

¼ ùS VV',(l \t--\ r\J

C..or-r¿s F°~¡vö -¼., -tf'QV\ ~✓-er.so.,]

THE Two PRIMARY PROPERTIES OF DILA TI ONS

When a dilation of a line segment AB not containing the center by a scale

factor of k produces A' B' then:

l. A'B'=k·AB -- --

2. A'B'II AB

Dc(A) = A'

Dc(B) = B'

e B B'

Name: ---------------- Date: --------

e DILATIONS 16-&Sworil\

MEASUREMENT AND CONSTRUCTION

1. Given the center point C, construct the dilated image of segment EF after a dilation by a scale factor of 3. Use only a compass and, straight edge. Label its image E'F'. Leave all marks.

z 7,5 CM

e 2. Verify using your compass that E' F' = 3 · EF in the diagram above. Leave your construction marks.

3. In the image below, point C lies on segment RS. User your ruler to help dilate RS using Cas a center with 1 -

a scale factor of - . Label the image R'S'. 2

R

S' s 4. Using your ruler, give the measurements of RS and R'S' in inches and verify that R'S'=..!_ RS.

2

Rs-a - '<!..w\ R'S'= 4,5°CVV\

5. Dilate MBC below using A as the center and a scale factor of 2. Leave all construction marks.

LlAß'e..'

6. Line segment MN is the image of CD after a dilation by a factor k " with a center at point A. Using your ruler, determine the value of k

to the nearest hundredth. Show the work that leads to your answer.

flrV1 = k .. "C.. d- I 7 ~ = k , 4,i.J('_WJ '-I. y ~ L-f. t.( cw

h-::o,toi PROBLEM SOLVING/REASONING

7. If JK is dilated by a factor of 5 with a center point not on JK to produce the image J' K', then which of the following is true?

Til) J'K'll7i and J'K'=5JK \ (2) J'K'J_JK and J'K'=5JK

(3) JKIIJ'K' and J'K'=]_JK 5

(4) JK J_J'K' and J'K'=]_JK 5

8. If RS is dilated by a factor of 4 with a center of R then which of the following is true about the segment joining point Swith its image S'? Hint: draw a picture of this one! j) (1) SS'=4RS and-SS'IIR-S

'X V 1 - - !i (2) SS'= - RS and SS' lies on top of RS

JV · 4

4') (3) SS'= ]_RS and SS' J_ RS

1 ~í(_4_) .. Sa..S.,,,.' =__:;,:R-S~a-nd---=S=S,,,..' -is_c_o_ll_in-e-ar_w_1_· th--=R=s7

9. Below, MBC has had the midpoints of sides AB and AC marked as D and E with DE drawn. Give a

dilation, both the center and the scale factor, that would map DE onto BC. Explain why this will work.

Center: r-:} Scale factor: f\ ::..;;}..

A

Explanatien: ·-

B e

Name: --------------- Date: -------

DILATIONS IN THE COORDINATE PLANE

In our first lesson on dilations, we learned about the fundamental concept of a dilation and its two very impmtant prnpcrtics.

THE Two PRIMARY PROPERTIES OF DILATIONS

When a dilation of a line segment AB not containing the center by a scale

factor of k produces A' B' then:

l. A 'B' = k · AB -- --

2. A'B'II AB

Dc(A) = A'·

Dc(B) = B'

e E B'

The coordinate plane gives us an excellent way to perform dilations and to verify these properties.

Exercise #1: The points A ( O, 2) , B ( -4, O) and C ( 3, 4) are plotted below. Find and plot their image points

after a dilation by a factor of 2 centered at the origin, O. Y

A(O, 2) ➔A'(o 11/) B(-4,0)➔ß' (-<?"

1o)

C(3,4)➔C' ( [o1 %)

¡ I

ci =l (~,. - ~, )Z- -+ {'11., - Y,) 2. 1--t-+-........+-+-,e-i-+-.........,_~ How can you verify that C' is twice the distance from the origin as C? l

11 •

~ ~ D = ~ (-; -o) 'l- +{ '-f -o) z.. l .=. J "9 ,. -t- i.J-L .= ~ l\ ·H ~ :-..f.Ts i-+-+---+-+-¡....;....¡.--+-+-; I j : ±j_-+-t-f ;: ~ ! ;

Dl' '=l b =( \ ~-o)J.. ,.j. c~-0)2- f ;- ~1o~+~z.1.::t ?,(ptiA 1 :::..~ :i ! I ._ç·-+-,'-Í-:--¡<-<I

-= 10

If I î ¡1 fl' I

I ., I ~ I , I

'/• -~~Lt- ., \ .L ' , I è," :5 :

Dilations from the origin are the most important in the coordinate plane. Their algebraic rule is extremely easy:

DILATIONS IN THE COORDINATE PLANE CENTERED AT THE ÜRIGTH

If ( x, y) is any point in the plane, its image after a dilation by a factor of k with a center at the origin has

coordinates given by ( lex, ky) .

Exercise #2: Which of the following is the image of ( -4, 1 O) after a reflection in the line y= x and then a

(_,¡_ ~ '-J) 'f:; ~ ( 'f I t) dilation centered at the origin with a scale factor of i? Q (,~\-u, f loV\ 2

(1)(-6,15)

@) (15, -6)]

(3) ( 6, -15)

(4) (-15,6)

(-4 , ¡ô) -> ( \ O I - 4)

(to •(l} 1 -\.I U)Ì g 15,-w[l

We can now more closely examine properties associated with dilations by using coordinate geometry tools.

Exercise #3: Segment AB shown below has endpoints at A(-1, 4) and B( 4, 2). y

(a) Plot the image of AB after a dilation by a scale factor of 3 with a center at the origin. Give its endpoints below.

(b) Find the slope of AB and A' B'. What do these slopes tell you about the segments?

Slope of A'B':

lo -17 +lo lm r-:2.. __ 3 == ïs - s

o I I i i ,.._ i I

""- I i

; i ' l ......... l I ! I I

I ! ! I l I ! ! ;

l. I '

-~~ß:_-_1_·~-,~!- !, ~-

( c) Find the lengths of AB and A' B' in sim lest radical form. Does this verify what we learned about dilations

previously? d:::- ('l-:z.-'l,)°z..+(Y-z..-'1,)2. Length of AB :

l)= r' '4 - - ,)'" .... (r;.-4)' :;:.r= 5'1- +- (_-~ L-

~ ~ ~5 +-4 = .JaÄ

Length of A' B' : , .D:: f ( 1;1 ---3) 7- + (<.,-12)-z.

= \ I':) 1- -\. (-&,) -Z. I

-- (" a.:J. 5 i-3,c.: =- Ja, lo I .: ff · {äP¡ =-Œ .ra:-;] So, we can now ven· usmg coordinates that dilations map a line segment to a parallel line segment of scaled length (assuming the center of dilation is not on the line segment).

Exercise #4: A line whose equation is 3 y+ Sx = 12 is dilated with a center at the origin, by a scale factor of _!_. 2

'/ \ v"kr~ rt l O i 'f)

Í I(_ ~_L ~ j (. O I ~-i) _::: (e, I 2-) '2- ?- 7'

~-== -5;< t_!t _ . _ ;~~'/í~ 3 3 3 --=-') ';j'= :.g.,.. + '-/ .¡,,.Sa.,,.,__. s 1. f@' f¡~ -¡ x 1-.;2. j

Exercise #5: Explain why dilating the line y= 3x by a scale factor of 2 about the origin would not change its

Write the equation of this line in slope-intercept form.

3j +5)(~,z.. -st -sx

equation.

Name: --------------- Date: ·,« ------- DILA TIO NS IN THE COORDINA TE PLANE

MEASUREMENT AND CONSTRUCTION " I

'.l " l. Given MBC shown, graph its image after a dilation .. centered at p the origin with a scale factor of 2. Give the coordinates of the ' I

' images of the vertices below. ... l\"o.. ti l-'-t, '-1) ~~z. '> A 1 (-i, 9) """" I ... I

ß (:~, -3) í3 I (-g-, -c.) ....... ~ ....

l(.=-l-? j 1, I.I '\ ~I ~X

' - :·Tn 1'1=2.> c.'(<?, -t.) ; I'- ' e l ~ -3) , __ ; I - I I

'l:l 'B=tffl 2. List all pairs of parallel line segments from problem 1. I: ! f"'I : ; i... 1~ I· ""'"'" Ae I\ A'a' I i ,- ~I i I !

I ,rs ! ~ ' ! I I l I . I ! . _! ¡ ! ! ,- - ~ I I ! I 1 - I : ; I • ' 1__L__ f ec " .. ß e;

PROBLEM SOLVING' y

3. In the following diagram, QR is the image of NP after· a

dilation centered at the origin with a scale factor of k. Which of the following would not be a correct calculation of k. - - Not.~

~:::: t>\Ó ~I) QR NP

'1J) OR OP

4 • N rn ...._.._ I i I

i-... I¡~'~+- '\,, f"'II Q~

""" ' I I .L I

~lii,,. 1..._:1- ¡. 1 I ~ ... ~~ i'""rH± . R ' ; I I . Ì,-r-+-¡-¡-¡

'-=i~-=-:- ~1.....,-¡=. o l l : : i I i . : 4¡-;-, I ;__¡_;_ .l._,

I • _:_,_L_¡ __ ;_j_¡ I -++··' I I _, I , : 1_J_: I :.

~X

~

~

• < ,. , 'l . ~..:. V~ , ' 4. If the line containing the points (-2, 10) and (7, 4 J is dilated by a factor of 2, then the slope of the new line

would be which of the following? d \q,-kJ + l,._.. S' \ op-e... re.MAJ~ +te_ (3) ~

3

(4) 2_ 3

5. Which of the following lines would ~have its equation change after a dilation.with a center at the origin?

(1) y=x+3

(2) y=I0

@) y=4x \ (3) x=6

6. If the line y@-3tm is dilated with a center at the origin and a scale factor of 4, which of the

following equations would describe its image?

~) j ? 12(x I S.) { (2) y - 28 = 3 (X+ 20) }

(3) y-28=12(x+20)

(<8, J' 28 3(x I 5) f

REASONING

_ • Xl ~. 'li 'fi 7. Line segmeryt AB has en_dpoints at A ( -4, 4) and B ( 8, 8).

(a) Find and plot its image after a dilation with a center at the ., '

origin and a scale factor of .!_·. Give the coordinates of A' and 2 'I 'l-t '1, A (-c..(,'-') ~& t.➔ A' (-o1, ~ J

'f..t, .'IA B ( r-, F') ~~ 'IL ::, ß \ (t.( I '-1)

B' below.

.,,,. I

!... , I,\ : , ~ f.¡Î ', 1- ....

J 31.,+ 4' -::-~t:)

-::-H. lt:)

ól .f'\b

unknown center point and unknown scaling facto

(a) Determine the scaling factor, k. Show how you a -s(-,,-,) e (3, '-f) E"(-1,-!>) F·(,,'l)

r a dilation with an I ¡ 7-j\r !~ l I j "Ì 'i.$.' I

' . ! I r. ; I , . V I\ ,

I ·h I I I ,,7.1~:- :v.1 "'I

rrived at your answer. I ¡ ~1 I i>'l I I Jj/'I ' ! I .. Id ! l ' 44/tJ .J/T I I ' ,.,,

e;F . ,-n,i::=<~~7='-ïf '\ - ~--- - - -- .. ßC - ;,.fO ✓ J_t__: __ : -- -~T !'\.

e, ,t t i I Î \

@i1 1 ! '•~--.¡!It

\ ; i! : -7 , ; I I I\ ~ ----...-,;------- --· -· .... __ , ... · ...... - ... - • IÌ....!,_j_ .. , \

I ; ! ' ---~Tl I t I\ I ·-r·--·- -. ~. ,-r7

A A-~ .. ; . --¡..-:-n-. - I I ; -~-· ·-1-17!{ ----~-- ·---a., -· _. ....... ....._._ ... .,

X

(b) Determine the coordinates of the center point graphically.

~ (5,-1)

Name: --------------- Date: ------- DILA TONS AND ANGLES

So far we have seen that dilations, whether in the Euclidean or coordinate plane, map segments to segments with the fullewlng properties:

THE Two PRIMARY PROPERTIES OF DILATIONS

When a dilation of a line segment AB not containing the center by a scale

factor of k produces A' B' then:

Dc(A) = A'

Dc(S) = B'

I. A I BI = k. AB

I -- -- 2. A'B'II AB U,,-----------------------------IUlllli!Al""F~!J!l7'l' "".lr..lüC·•,•·•œ:·•r~•'l'!~"'•i!!· !1.'.,~t!:9· 31:!Œl-- e B B'

In this lesson we are going to see one additional property that is extremely important.

Exercise #1: In the following diagram, L.BAC has been dilated with a center of D and a scale factor of 2.

(a) List all parallel rays shown in the diagram. ►i-ll 11 /ft~ A~ J\ A1ß1

(b) Using your protractor, what are the measures of L.1 and L.3?

mLI=":}5°

D •

B' (c) Give a reason based on the second property of dilations for why the angle measure doesn't change in this

d_ila~on. Refe1>to the dlagralll.abo}e and use proper terminology regarding parallel lines. ~V\ AC. It fl 'e. 1 1 Ai I I '4181 G:\Je..Y\ ìi,..detoce I b'{ subs\.;~/¡~

Äl I I A. e ---~ LI :k L '2. c..o rn:.spol'dr' 'ô Q "'-j k, _ _s- L I ~ L.3 ¡¡f¡ I I A I ß ~ ---~ LJ ::J£ L3 {!.t1r(ö po....d-1 'i ~ ~7 (-eJ' ...

THIRD IMPORT ANT PROPERTY OF DILATIONS

Dilations are angle preserving. In other words, the angles of a geometric object do not cha tge when dilated.

Exercise #2: If !!JJEF is dilated by a factor of 5, which of the following statements would. t.:.: true?

1 ·1:) ·,\lt, ~'b~s bö ~ ', ( 1) mL:D' = S ,,~D (~.;;,.;~1-;,,,.,.nL-r-E~-5-;mn-rL~t/' Ì) 1 C'..,l,,,o.,wj .,,__ .+t,_, QV\J \~s (2) DE~½D'E' {'I) EF 5E'fi'' ~'5Á- s;cks_. __

E' F'

Exercise #3: In the diagram below, tiE 'F 'G' is the image of tiEFG after a dilation of an unknown scale factor E' with an unknown center point.

(a) Using tracing paper, verify that the angles of MFG are congruent to the angles of 4]i: 'F 'G'.

I

(b) Graphically determine the location of the center of the dilation. Mark this point D. Leave all construction marks.

'

G'

(c) Using your ruler to measure lengths, calculate the scale factor to the nearest tenth. Verify it with at least two sets of sides. There may be errors involved due to rounding side lengths.

r:.' F • - ~ ~ :::- r-4 €:vV :::. a,..~ -= I o (e • r.:;.; - (J,o CM ô\Ö \,u E'F.::: ,.rev\,\ -- ' ..... . . - FG'= '5_$Q..,'M ffi .= 3. ¿, C\.v\

Exercise #4: -In the diagram of MBC below, D and E have been located on AB and AC such that DE li BC. 8

(a) Give a dilation that will map BC onto DE. Specify both the center and scale factor of the dilation. Justify. b E ~

C~ ~ A ~• - \, tJew-~ ~ "" ~ ~ -o\<J - ßC \_

v:,.

~iç w.\,\ wdJ!- b}c. 'ß · c.' :::: I?E. QJ A --~~- e ~ w\ \\ he., pa.tQ\leJ ·\-ö ife s-o ~os-l 1~€. c,111 -1-öf ~ })£.

(b) If MBC was transformed using the dilation specified in ( a), explain why its image M 'B 'C' would be

congruent to MDE. fJ w~ Mll.f> --b p¡ 1 B Woù Id "'1,(;\. f ·-le, Ì) Cl,v\.d

C.. Wc0\A wictf 42> E. $ b 4\l ~Q \J ert,·~t of' DA' ife. w<H.>fd fltt /} ÔV\ iADE

(c) If AD= 10, AB= 15, BC= 12, a~ AE = 14, then algebraically determine the lengths of DE and AC.

\ IC I'\ l ~ ~ -::._!5 := Í, Ç lo/ Ll 10

. )9-- ' È J4 a.' c.

DE-=--~ ~ 15be.: no ,~ 15 /5 - 15

DE.=-î

ftl _ 12. _, <? f\l'- ~ l <a'ô -¡rr - ~ -1 B ~

4¿.:=- ;J\

Name: --------------- Date: -------

DILATIONS AND ANGLES

MEASUREMENT AND CONSTRUCTION

l. Given LCDE shown below, construct its image, LC'D'E' after a dilation using F as the center and a scale factor of 2. Leave all construction marks.

2. Using your protractor, find the measure of both of the following a 1.

(a) mLCDE = (;Oº

(b) mLC'D'E' = (d:)º

í'G VVlÖL; v~ +Lt,_

tk- Q_,vcj le_ PROBLEM SOLVING

p' 3. Why isn't the measure of LC'D'E' twice the measure of LCDE in #2 above?

1o ¡ c. dila._ +tò "

4. If rectangle RSTU is dilated with a scale factor of _!__ with a center at R then which of (he following is the 2 '

value of mLS'T'U'?

(1) 45º

0 (3) 120º

(JTÍJ~~,

(4) 180º

5. In rhombus ABCD shown, mLA = 140º and diagonal BD has been drawn.

If side BC is dilated with D as the center and a using a scale factor of _!__ , 2

then which of the following would be the measure of LDB 'C'?

( 1) 1 Oº

~2ò~1 (3) 40º o

A --=·=- =-, ~~---,r--7 B

w \..._.c. I~ iJ,.:v1..3)e._,

LB= Lf{i}

(4) 70º

6. In isosceles triangle ABC it is known that AB= AC. If MBC is dilated using B as the center point, then which of the following could be an incorrect.statement? Draw a good diagram!

/c1) LBA e :.: LBA 'e ' ., ;:i

~) LABO .. LA'C"'B¡

f3) LC'A'B ~ LABC] 04) LBCA ~ z':A L-BG\

7. In the following diagram, D and E lie on sides AB and AC such that DE li BC as marked. It is known that AD= 18 and DB= 12.

A

(a) Give a dilation that would map BC onto DE. State both the center of the dilation and the scaling factor.

(b) If BC = 15 and AC= 20 , then find the perimeter of MDE . Show how you arrived 2-r ycur answer.

8. Given two concentric circles ( circles that have the same center point), with the inner circle having a radius of ,¡ and the outer one having a radius of

r2 as shown, give a dilation that would map the inner circle onto the outer circle. Specify both the center of dilation and the scale factor.

REASONING

9. Explain why two perpendicular lines will remain perpendicular if they are both dilated 1:iy the same factor and using the same center.

Name: ---------------- Date: -------- SIMILARITY

COMMON CORE GEOMETRY

We have now see that when a figure is dilated that all side lengths are scaled according to the scaling factor and tñe angles are preserved. ln this sense, we say ûtat the two images have the sa:me .dta1:,e bt1t different sizes. Two shapes that share such properties are known as similar figures.

SIMILAR FIGURES

Two geometric figures are similar, ~ , if all pairs of corresponding sides are propordoual and the corresponding angles are of equal measure.

Exercise #1: In the diagram below, we know that MEC~ WEF.

(a) Use tracing paper to verify that all corresponding angle pairs between the two triangles are the same measure.

(b) Solve for the missing side lengths of both triangles. Show your work or explain your reasoning. ,-I 'i --d- -, FE - -

~

e

F

1-

D ~ E

( c) Give a dilation of MEC that would guarantee that its image, M 'E' C', is congruent :e, !iDEF . How do youknowitiscongruent? ~~ \6.-t<d b /i'8l- v-'~ a., .S(!.4'4t %d-or" ~ \-l::. ~ Q..""-'{ t.e.\l\.~,.

The last example gives rise to the technical definition of similarity. In a way that is very sire.rar te congruence, we will define similarity using transformations.

SIMILARITY TRANSFORMATIONS

Two geometric figures are similar if there exists a similarity transformation that wil. map one figure onto the other. This transformation is either a dilation alone or a dilation in combination v1è. one or more rigid motions. (

L.:.----~;:¡,l:~¡¡¡;;~~~~--~~~~~~~.U..a.J~~l..:l~~~=~''C:=-••'1'1:ó===·c=--__, Exercise #2: Given that MEC and WEF below are similar, with congruent angles marked, g.ve a similarity transformation that would map MBC onto !::JJEF. Using tracing paper to help. Be a~ spec.fic ts possible.

(J) µ..o-~ A «ec a.'bciA- ß

Ú:) Tr().,Y\s\o.-~ .\e?

e

f 5D ~ B

Exercise #3: In the following diagram, it is known that BCN and ACM and MBC ~ íJ.MNC. Give a similarity transformation that would map MBC onto ßMNC. Use tracing paper to help visualize the rigid motions.

0 \2n\,,),e {!) b·, lo.-\~

Exercise #4: Given MBC ~ WEF on the grid shown:

(a) What is the scaling factor needed to dilate MBC so that it is congruent to WEF ? Show how you arrived at your answer.

"',t,vJ . •-'l-,,. /_ r,J . !::-1.1 - (£) 1 ~ ò\Ó= '"7-íé, -7f:: óV

é ~ .. ! ,-,-....-,---r-,,-,-.,--,-~-. -,-1 ....,,._--:- ' r 1 , , ; •

1---+- +-+--+-+-,,--l----l:'~~77-ÎT-;---;- ...;.--+ ----+--i

1---+-+-+--+--11-t-+-+--+' ] ! !_!_; +-+--+-+-+-i 1---+- +-+--t-t--+-+-+-+ t, '

t-+-+-t- --+-+--+-t-+-+- ! t-+-+-t- --+--+ --+-t-+-+-ì_

I H-+-A'P'II~.._•=¡~--¡-- ·-i

" !

(b) Specify a similarity transformation that would map MBC onto WEF.

/"':-,.. \tn,~~ ~ A A \"'.2.,. ~ \ ~ ~H. e>- "e,.. \-t\lc~ ß fk4-fS -fo £ @ Re~ ~A. ß1¿ &l'Ct>S& CJCîoss ·x.::3 Ctrr e,~) ® \:); ~ ,1 4 I\ ß .1 e, ì l IP~-\¼ C-LL-t-+<.r E "--vtC\ q,_ Sl!O ~

.ft_~ of K-=i.S- Exercise #5: In the diagram below A, B, and C are collinear as well as A, D, Md Jf. I(

I I

' ; I __ · __ , ~, : ¡ ;

l---+-+-t--t-+-t-11 ~,;--:.- =! 1---+-+-t--t---;t--+--'ti.~-- ~\ ~---

......... ~-~¡~¡ I~~-•.,_; .-. .. .., . ...,~-----•4•-

~ xnown that

BDIICE. J.

(a) Give a dilation that would map MBD onto MCE.

\'1::: ~e,w - A~ ;:: _fl§ :: ~ o \e\ - A ß A 'D ß D

(b) Circle the proportion that is incorrect based on MBD ~ M CE .

~~C = CE AB BD

~~(mLA = mLA mLABD mLACE

- l

( ... ) AD AC 11l-=-

AE A,B

~ " .. tot 1nLAEr n· , ,t, 1 ·B (iv) ,._, = _:._":_-~-- mLA CF rr:./.A.FD

Name: -------------- Date:

SIMILARITY

__ PROBLEM-80LVING"-------------------------------

1. In the diagram below, MBC ~ WEF with side lengths as marked. Set.up proportions a:::.d selve them to find the missing sides of WEF . Show the work that leads to your answers.

o

\'···-~ ... , \ ~, . 8)

ólS \ ",,,

\_u,·a.~=~ •~•·"'~~-~~ B 14 C E 25 F

_llf fl_C _ ~ 2i' -~ = J-t 7o(JÇJ ~1/'Í_DF

/)F - f:~ -'/ DF J s ?fP:::- I~»,., 2. In the diagram below, t:..MNP has been dilated by an unknown scaling factor centered ~~~ .11:;;o produce

l.lMQR. Which of the following is an incorrect statement? R

p /f I 6 i I ' í i _________ J

M N Q

"3,5(11.::,J -::: I 4 i'.:> E. "350:: 14 DE ri ¡L} *

as~De:

A

10

(I) t:MNP ~ l.lMQR

(2) mLMNP = mLMQR mLMPN mLMRQ

(3) MR= RQ MP NP

(4) MN =MR MQ MP

3. In the following diagram, AE and BD intersect at C such that MBC ~MDC. vvh1;c!ï of the sets of lengths below are possible, given this information?

~ AC=lO, AB=6, CE=?, andDE=3

î_®RC=12, CD= 8, AC=9, and CE= 6

(3) AB=20,DE=I0,DC=7 andBD=I4

(4) AB=I8,BC=24,DE=I0, andDC=16

A

B~_j-Oßl ~~,-u,

4. In the diagram shown, MBC~MDE. If AB=I2,BD=8, AC=18, and

10 (p --- ~ ·-- -, 3

CE= í.-2, rnen what dilation

scaling factor below would map MBC to MDE given A as the center?

(1) k = ¾ F k = TI } -J2 ~ J.'.íi:,----;.\ /~ i - \1..- B.

A ·'

(2) k = '!:_ 3

(4) k =i 5

5. Given MBC shown below. Using only a compass and straightedge, dilate MBC using A as the center and a scaling factor of 2. Explain why MB' C' must be similar to MBC.

REASONING

6. In the diagram shown, ~EF .

Describe a sequence of transformations that would map MBC onto WEF . Be as specific as possible. Use tracing paper if needed. B

·, (<U¡1s\o...,'.\..~ I+ -to v¾º~ {o b J2 efle,c..\:'\i o~.u- ~ t.--~~ ~J~ b: \ °'-:\i Cl,'\ Ls hf~ V\ \(,.- ')

y In the diagram below, WEF is the image of MBC after a dilation

centered at the origin with a scale factor of ..!_ . 2

(a) If AD and EB were drawn, explain why quadrilateral ADEE would be a trapezoid.

(b) If MBC was first translated three units left and one unit down so that C corresponded with F, would the scale factor needed to map M'B'C' onto WEF change? Why or why not

Name: ..... tk----~..,,._ Period: __ Date: _ Unit 4 SÏmiíJrity- Review

Caleb sketched the following diagram on graph paper. He dilated points Band C from point O. Answer the following questions based on his drawing:

1. What is the scale factor k? Show your work.

~ ôß', . \ ----- - _L_ - - Oß - (e - 3

..

2. Verify the scale factor with a different set of segments.

3. Which segments are parallel? How do you know?

Be. 11 8 'c..• - bo¼ ~Q.\l~ .\-\.¿_ s~~ s\or-t- ' (_ 6)

4. Are< DCB and < OC'B' congruent? How do you know?

c. .. , J'" ~- bl e ll~f-ts Corre~ 'f>ON:Ü~ "'-"'j\-t.

p

S) Looking at the triangles in the figure on the right: a) Are the two triangles similar? b) What is the length of Q 7? e) If PTis 15 cm, what is the length of R7?

( ~) ~¿Ç l -\\.t_, ('rJ) s~ =-~

Q.r ?~

9cm 3 cm,

6) In the following diagram, AE and BD intersect at C such that ó.ABC~ó.EDC. Find the missing lengths on the triangles. AC = 12, AB= 9, CD= 5, and DE = 3

D

B

=::ì

- -

t.-\ E :::. 3 (&~) -- 3 -3

' ,;-: ß~""\ 5(.¿, ;;: qec E) - ---- 1\ q

l~:: ~~