11.2 Proving Figures are Similar Using Transformations

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1 Name lass ate 11. Proving igures are Similar Using Transformations ssential Question: How can similarit transformations be used to show two figures are similar? esource ocker plore onfirming Similarit similarit transformation is a transformation in which an image has the same shape as its pre-image. Similarit transformations include reflections, translations, rotations, and dilations. Two plane figures are similar if and onl if one figure can be mapped to the other through one or more similarit transformations. grid shows a map of the cit park. Use tracing paper to confirm that the park elements are similar. Trace patio HG. Turn the paper so that patio HG is mapped onto patio MON. escribe the transformation. What does this confirm about the patios? H -8 - N 0 G 8 O Houghton Mifflin Harcourt Publishing ompan - N M O 0 M - Trace statues and MNO. old the paper so that statue is mapped onto statue MNO. escribe the transformation. What does this confirm about the statues? Module esson

2 escribe the transformation ou can use to map vertices of garden ST to corresponding vertices of garden. What does this confirm about the gardens? S T eflect 1. ook back at all the steps. Were an of the images congruent to the pre-images? If so, what tpes of similarit transformations were performed with these figures? What does this tell ou about the relationship between similar and congruent figures?. If two figures are similar, can ou conclude that corresponding angles are congruent? Wh or wh not? plain 1 etermining If igures are Similar You can represent dilations using the coordinate notation (, ) (k, k), where k is the scale factor and the center of dilation is the origin. If 0 < k < 1, the dilation is a reduction. If k > 1, the dilation is an enlargement. ample 1-0 T Z etermine whether the two figures are similar using similarit transformations. plain. ST and XYZ To X S Y map ST onto XYZ, there must be some factor k that dilates ST. Pre-image Image (0, 1) X (0, 3) S (1, -1) Y (3, -3) T (-1, -1) Z (-3, -3) Houghton Mifflin Harcourt Publishing ompan Module esson

3 You can see that each coordinate of the pre-image is multiplied b 3 to get the image, so this is a dilation with scale factor 3. Therefore, ST can be mapped onto XYZ b a dilation with center at the origin, which is represented b the coordinate notation (, ) (3, 3). dilation is a similarit transformation, so ST is similar to XYZ. To PQS and WXYZ 10 X(, 9) Y(1, 9) 8 W(, ) Z(1, ) Q(, ) (, ) P(, ) S(, ) map PQS onto WXYZ, there must be some factor k that enlarges PQS. Pre-image P (, ) Image Q (, ) X (, 9) Z (1, ) ind each distance: PQ =, Q =, WX =, and = 7 If kpq = WX, then k =. However. Q = / XY. No value of k can be determined that will map PQS to WXYZ. So, the figures are/are not similar. Your Turn etermine whether the two figures are similar using similarit transformations. plain. 3. MNO and GH 1 Houghton Mifflin Harcourt Publishing ompan (, 1) (8, 1) O(1, ) N(, ) G(, ) H(8, ) (1, ) M(, ) Module esson

4 . and MNP. and TUV - 0 M - U 0 T V N P plain inding a Sequence of Similarit Transformations In order for two figures to be similar, there has to be some sequence of similarit transformations that maps one figure to the other. Sometimes there will be a single similarit transformation in the sequence. Sometimes ou must identif more than one transformation to describe a mapping. ample ind a sequence of similarit transformations that maps the first figure to the second figure. Write the coordinate notation for each transformation. to HG - 0 G H Since HG is smaller than, the scale factor k of the dilation must be between 0 and 1. The length of _ is and the length of _ is ; therefore, the scale factor is 1. Write the new coordinates after the dilation: Original oordinates (1, ) (, ) (, ) (, ) oordinates after dilation k = 1 ( 1, 3 ) (, 3 ) (-1, 1) (1, 1) translation right units and down 3 units completes the mapping. oordinates after dilation ( 1_, 3 ) ( _, 3 ) oordinates after translation ( +, - 3) (, 0 ) ( (-1, 1) (1, 1) 9, 0 ) G (1, ) H (3, ) Houghton Mifflin Harcourt Publishing ompan The coordinates after translation are the same as the coordinates of GH, so ou can map to HG b the dilation (, ) 1 (, 1 ) followed b a translation (, ) ( +, - 3). Module esson

5 to PQ P Q 8 You can map to PQ with a reflection across the -ais followed b a followed b a counterclockwise rotation about the origin. eflection: (, ) (, -) : (, ) ( ) (, ) ( ) counterclockwise rotation: eflect. Using the figure in ample 3, describe a single dilation that maps to HG. 7. Using the figure in ample 3, describe a different sequence of transformations that will map to PQ. Houghton Mifflin Harcourt Publishing ompan Your Turn or each pair of similar figures, find a sequence of similarit transformations that maps one figure to the other. Use coordinate notation to describe the transformations. 8. PQS to TUVW -8 - V W U T 0 Q P S 8 Module esson

6 9. to escribe a sequence of similarit transformations that maps MN to VWXYZ. M N W V X Z Y plain 3 Proving ll ircles re Similar You can use the definition of similarit to prove theorems about figures. ircle Similarit Theorem ll circles are similar. ample 3 Prove the ircle Similarit Theorem. Given: ircle with center and radius r. ircle with center and radius s. r s Houghton Mifflin Harcourt Publishing ompan Prove: ircle is similar to circle. To prove similarit, ou must show that there is a sequence of similarit transformations that maps circle to circle. Module 11 9 esson

7 Start b transforming circle with a along the vector. s r r ircle ' Through this, the image of point is. et the image of circle be circle ʹ. The center of circle ʹ coincides with point. Transform circle ʹ with the dilation with center of dilation ircle ʹ is made up of all the points at distance from point. and scale factor s_ r. fter the dilation, the image of circle ʹ will consist of all the points at distance from point. These are the same points that form circle. Therefore, the followed b the dilation maps circle to circle. ecause and dilations are, ou can conclude that circle is to circle. eflect 11. an ou show that circle and circle are similar through another sequence of similarit transformations? plain. 1. iscussion Is it possible that circle and circle are congruent? If so, does the proof of the similarit of the circles still work? plain. Houghton Mifflin Harcourt Publishing ompan laborate 13. Translations, reflections, and rotations are rigid motions. What unique characteristic keeps dilations from being considered a rigid motion? 1. ssential Question heck-in Two squares in the coordinate plane have horizontal and vertical sides. plain how the are similar using similarit transformations. Module esson

8 valuate: Homework and Practice In ercises 1, determine if the two figures are similar using similarit transformations. plain. 1. GH and Online Homework Hints and Help tra Practice Pre-Image Image - H 0 G -. PQ and STU S - 0 Q T U 3. MN and PQS -8 - Q P 0 M 8 Houghton Mifflin Harcourt Publishing ompan S N Module 11 9 esson

9 . UVW and GHI U W - 0 H G V I or the pair of similar figures in each of ercises 10, find a sequence of similarit transformations that maps one figure to the other. Provide the coordinate notation for each transformation.. Map to PQ.. Map to GH. - 0 Q P H G 7. Map to. 8. Map to MN. Houghton Mifflin Harcourt Publishing ompan N M Module 11 9 esson

10 9. Map to M. 10. Map to PQ -8-0 M 8 - Q 0 P omplete the proof. 11. Given: Square with side length. Square GH with side length. H Prove: Square is similar to square GH. G Houghton Mifflin Harcourt Publishing ompan Module 11 9 esson

11 1. Given: quilateral with side length j. quilateral PQ with side length p P Prove: is similar to PQ. j p Q 13. Given: with = c, = a, = b c XYZ with YZ =, XY = a, XZ = b a Y Prove: is similar to XYZ. c a c a b X b a Z Houghton Mifflin Harcourt Publishing ompan Image redits: ill Hustace/orbis 1. The dimensions of a standard tennis court are 3 feet 78 feet with a net that is 3 feet high in the center. The court is modified for plaers aged 10 and under such that the dimensions are 7 feet 0 feet, and the same net is used. Use similarit to determine if the modified court is similar to the standard court. Module esson

12 1. epresent eal-world Problems scuba flag is used to indicate there is a diver below. In North merica, scuba flags are red with a white stripe from the upper left corner to the lower right corner. ustif the triangles formed on the scuba flag are similar triangles. 1. The most common picture size is inches inches. Other common pictures sizes (in inches) are 7, 8 10, 9 1, 11 1, 1 18, and 1 0. a. re an of these picture sizes similar? plain using similarit transformations. b. What does our conclusion indicate about resizing pictures? 17. Nicole wants to know the height of the snow sculpture but it is too tall to measure. Nicole measured the shadow of the snow sculpture's highest point to be 10 feet long. t the same time of da Nicole's shadow was 0 inches long. If Nicole is feet tall, what is the height of the snow sculpture? 18. Which of the following is a dilation?. (, ) (, 3). (, ) (3, -). (, ) (3, 3). (, ) (, - 3). (, ) ( - 3, - 3) 19. What is not preserved under dilation? Select all that appl.. ngle measure. etweenness. ollinearit. istance. Proportionalit Houghton Mifflin Harcourt Publishing ompan Image redits: (t) huck Wagner/Shutterstock; (b) ave eede/esign Pics Inc./lam Module esson

13 H.O.T. ocus on Higher Order Thinking 0. nalze elationships onsider the transformations below. I. Translation II. eflection III. otation IV. ilation a. Which transformations preserve distance? b. Which transformations preserve angle measure? c. Use our knowledge of rigid transformations to compare and contrast congruenc and similarit. ustif easoning or ercises 1 3, use the figure shown. etermine whether the given assumptions are enough to prove that the two triangles are similar. Write the correct correspondence of the vertices. If the two triangles must be similar, describe a sequence of similarit transformations that maps one triangle to the other. If the triangles are not necessaril similar, eplain wh. 1. The lengths X, X, X, and X satisf the equation X X = X X. X. ines and are parallel. Houghton Mifflin Harcourt Publishing ompan 3. X is congruent to X. Module esson

14 esson Performance Task nswer the following questions about the dartboard pictured here. 1. re the circles similar? plain, using the concept of a dilation in our eplanation.. You throw a dart and it sticks in a random location on the board. What is the probabilit that it sticks in ircle? ircle? ircle? ircle? plain how ou found our answers. Houghton Mifflin Harcourt Publishing ompan Module esson