11.2 Proving Figures are Similar Using Transformations
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1 Name lass ate 11. Poving igues ae Simila Using Tansfomations ssential Question: How can similait tansfomations be used to show two figues ae simila? esouce ocke ploe onfiming Similait similait tansfomation is a tansfomation in which an image has the same shape as its pe-image. Similait tansfomations include eflections, tanslations, otations, and dilations. Two plane figues ae simila if and onl if one figue can be mapped to the othe though one o moe similait tansfomations. gid shows a map of the cit pak. Use tacing pape to confim that the pak elements ae simila. Tace patio HG. Tun the pape so that patio HG is mapped onto patio ON. escibe the tansfomation. What does this confim about the patios? H - -N G O - Houghton ifflin Hacout Publishing ompan otation of 10 aound the oigin; The fountains ae simila. - N - O - J Tace statues and JNO. old the pape so that statue is mapped onto statue JNO. escibe the tansfomation. What does this confim about the statues? eflection acoss the -ais; The figues ae simila. odule 11 7 esson O NOT IT--hanges must be made though "ile info" oectione=n-;- ate
2 escibe the tansfomation ou can use to map vetices of gaden ST to coesponding vetices of gaden. What does this confim about the gadens? S T dilation with scale facto 1_ ; The gadens ae simila. eflect 1. ook back at all the steps. Wee an of the images conguent to the pe-images? If so, what tpes of similait tansfomations wee pefomed with these figues? What does this tell ou about the elationship between simila and conguent figues? The two figues in Steps and ae conguent to each othe. The tpes of tansfomations that wee pefomed with these figues wee otations and eflections, which ae igid motions. So conguent figues ae also simila figues.. If two figues ae simila, can ou conclude that coesponding angles ae conguent? Wh o wh not? Yes, the coesponding angles ae conguent because igid motions and dilations peseve angle measues. plain 1 etemining If igues ae Simila You can epesent dilations using the coodinate notation (, ) (k, k), whee k is the scale facto and the cente of dilation is the oigin. If 0 < k < 1, the dilation is a eduction. If k > 1, the dilation is an enlagement. ample 1-0 T Z etemine whethe the two figues ae simila using similait tansfomations. plain. ST and XYZ To X S Y map ST onto XYZ, thee must be some facto k that dilates ST. Pe-image Image (0, 1) X (0, 3) S (1, -1) Y (3, -3) T (-1, -1) Z (-3, -3) Houghton ifflin Hacout Publishing ompan odule 11 esson
3 You can see that each coodinate of the pe-image is multiplied b 3 to get the image, so this is a dilation with scale facto 3. Theefoe, ST can be mapped onto XYZ b a dilation with cente at the oigin, which is epesented b the coodinate notation (, ) (3, 3). dilation is a similait tansfomation, so ST is simila to XYZ. To PQS and WXYZ 10 X(, 9) Y(1, 9) W(, ) Z(1, ) Q(, ) (, ) P(, ) S(, ) map PQS onto WXYZ, thee must be some facto k that enlages PQS. Pe-image Image P (, ) W (, ) Q (, ) X (, 9) (, ) Y (1, 9) S (, ) Z (1, ) ind each distance: PQ =, Q =, WX =, and XY = 7 If kpq = WX, then k =. Howeve. Q = / XY. No value of k can be detemined that will map PQS to WXYZ. So, the figues ae/ae not simila. You Tun etemine whethe the two figues ae simila using similait tansfomations. plain. 3. NO and GHJ 1 Houghton ifflin Hacout Publishing ompan 1 10 (, 1) O(1, ) N(, ) G(, ) (1, ) (, ) J(, 1) H(, ) 0 10 Yes, ou can use a dilation with scale facto and cente at the oigin to map NO onto GHJ. The figues ae simila because a dilation is a similait tansfomation. odule 11 9 esson
4 . J and NP. and TUV J - 0 T - U 0 V N P No, the angles ae diffeent. J and NP ae not simila figues. Yes, thee is a dilation centeed at point with a scale facto of. is simila to TUV. plain inding a Sequence of Similait Tansfomations In ode fo two figues to be simila, thee has to be some sequence of similait tansfomations that maps one figue to the othe. Sometimes thee will be a single similait tansfomation in the sequence. Sometimes ou must identif moe than one tansfomation to descibe a mapping. ample ind a sequence of similait tansfomations that maps the fist figue to the second figue. Wite the coodinate notation fo each tansfomation. to HG - 0 G H Since HG is smalle than, the scale facto k of the dilation must be between 0 and 1. The length of _ is and the length of _ is ; theefoe, the scale facto is. Wite the new coodinates afte the dilation: Oiginal oodinates (1, ) (, ) (-, ) (, ) oodinates afte dilation k = (, 3 ) (, 3 ) (-1, 1) (1, 1) tanslation ight units and down 3 units completes the mapping. oodinates afte dilation ( 1_, 3 ) ( _, 3 ) (-1, 1) (1, 1) oodinates afte tanslation ( +, - 3) (, 0 ) ( 9, 0 ) G (1, -) H (3, -) Houghton ifflin Hacout Publishing ompan The coodinates afte tanslation ae the same as the coodinates of GH, so ou can map to HG b the dilation (, ) (, ) followed b a tanslation (, ) ( +, - 3). odule esson
5 J to PQ J P Q You can map J to PQ with a eflection acoss the -ais followed b a dilation followed b a 90 counteclockwise otation about the oigin. eflection: (, ) ilation (, -) : (, ) ( 3, 3 ) (, ) ( -, ) 90 counteclockwise otation: eflect. Using the figue in ample 3, descibe a single dilation that maps to HG. connecting the coesponding vetices, ou can identif (, -) as the cente of dilation fo a dilation with scale facto that maps to HG. 7. Using the figue in ample 3, descibe a diffeent sequence of tansfomations that will map J to PQ. eflect J acoss the -ais. Then dilate with cente at the oigin and scale facto 3. Then otate 90 clockwise aound the oigin. Houghton ifflin Hacout Publishing ompan You Tun o each pai of simila figues, find a sequence of similait tansfomations that maps one figue to the othe. Use coodinate notation to descibe the tansfomations.. PQS to TUVW - - V W - U T 0 Q P S You can map PQS to TUVW b a eflection followed b a dilation. eflection: (, ) (-, ) ollowed b, 3 ) ilation: (, ) ( 3 odule esson
6 9. to You can map to b a otation about the oigin 10 followed b a dilation followed b a tanslation. otation: (, ) (-, -) ollowed b ilation: (, ) ( 3_, 3_ ) ollowed b Tanslation: (, ) ( - 3, + 1.) escibe a sequence of similait tansfomations that maps JN to VWXYZ. J N W V X Z Y Tanslate JN ight 7 units so that J maps to V. eflect JN acoss JN. ilate JN with cente J and scale facto 1_. plain 3 Poving ll icles e Simila You can use the definition of similait to pove theoems about figues. icle Similait Theoem ll cicles ae simila. ample 3 Pove the icle Similait Theoem. Given: icle with cente and adius. icle with cente and adius s. s Houghton ifflin Hacout Publishing ompan Pove: icle is simila to cicle. To pove similait, ou must show that thee is a sequence of similait tansfomations that maps cicle to cicle. odule 11 9 esson
7 Stat b tansfoming cicle with a tanslation along the vecto. s icle ' tanslation point Though this, the image of point is. et the image of cicle be cicle ʹ. The cente of cicle ʹ coincides with point. Tansfom cicle ʹ with the dilation with cente of dilation icle ʹ is made up of all the points at distance fom point. and scale facto s_. fte the dilation, the image of cicle ʹ will consist of all the points at distance fom point. tanslation tanslations ou can conclude that cicle is simila to cicle. s_ = s These ae the same points that fom cicle. Theefoe, the followed b the dilation similait tansfomations maps cicle to cicle. ecause and dilations ae, eflect 11. an ou show that cicle and cicle ae simila though anothe sequence of similait tansfomations? plain. Yes, can eflect cicle acoss the pependicula bisecto of _, mapping point to point. Then, follow the same steps fo dilation. Houghton ifflin Hacout Publishing ompan 1. iscussion Is it possible that cicle and cicle ae conguent? If so, does the poof of the similait of the cicles still wok? plain. Yes, if the atio of each cicle is the same positive value. The poof still woks, because the atio of s to is 1, so the dilation does not affect the size of the cicle. laboate 13. Tanslations, eflections, and otations ae igid motions. What unique chaacteistic keeps dilations fom being consideed a igid motion? Unlike the othe tansfomations, dilations don't peseve distance, meaning the length of the sides will not sta the same between the pe-image and its image. The lengths of coesponding sides will be popotional accoding to the scale facto used. 1. ssential Question heck-in Two squaes in the coodinate plane have hoizontal and vetical sides. plain how the ae simila using similait tansfomations. Possible answe: tanslate the bottom left vete of one squae to the othe squae. Then dilate the fist squae b the atio of the side lengths. odule esson
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