9-1. Translations. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary

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1 9- Translations Vocabular Review. Underline the correct word to complete the sentence. If two triangles are congruent, corresponding angle measures are the same/ different and corresponding side lengths are the same/ different.. transformation is a change in form or appearance. Which picture does not show a transformation of the soccer ball?... Vocabular uilder translation (noun) truh anz la shuh n Main Idea: translation describes how a figure in a coordinate plane is slid from one place to another. Eample: Each point in $DEF was moved 5 units to the right. $D'E'F' is a translation of $DEF F O D D E F E Use Your Vocabular 3. Tell whether the pair of figures shows a translation. Write es or no. es no. In what direction is the translation of figure D? left 5. How man units has figure D been translated? units 5 3 D O hapter 9 6 D opright b Pearson Education, Inc. or its affiliates. ll Rights Reserved. Related Words: transformation, slide, preimage, image

2 Ke oncept Translation translation is a transformation that maps all points of a figure the same distance in the same direction. You write the translation that maps onto ''' as T() '''. translation is a rigid motion with the following properties: If T() ''', then ' ' ' '', '', '' m m', m m', m m' ircle the correct word in each statement. 7. Translations move some / all points in a figure the same distance, in the same direction. 8. Translations change / preserve side lengths of the figure. 9. Translations change / preserve angle measures of the figure. Problem Identifing a Rigid Motion Transformation Transformation opright b Pearson Education, Inc. or its affiliates. ll Rights Reserved. Image Preimage Image Preimage Got It? Does the transformation appear to be a rigid motion? 0. ompare the corresponding sides of the preimage and image. re the equal in length? Transformation : es no Transformation : es no. ompare the corresponding angles of the preimage and image. re the equal in measure? Transformation : es no Transformation : es no. rigid motion preserves side lengths and angles measures of the preimage. Transformation : is / is not a rigid motion. Transformation : is / is not a rigid motion. 7 Lesson 9-

3 Problem Naming Images and orresponding Parts In the diagram, NID SUP. Got It? What are the images of I and point D? What are the pairs of corresponding sides? S U N I P 3. Use the position of the letters in the transformation statement NID SUP. a. ngle I is in the nd position in NID. The angle in the nd position for SUP is U. b. Point D is in the 3rd position in NID. The point in the 3rd position for SUP is point P.. Name the three pairs of corresponding sides using NID SUP. NI and SU ID and UP ND and SP D Problem 3 Finding the Image of a Translation Got It? What are the images of the vertices of T, ()? Graph the image of. 5. Identif the coordinates of the vertices of. (, ) (, ) ( 0, ) 6. Describe the translation rule, T,. dd Subtract to each -value. from each -value. 7. Use the rule to find the coordinates of the vertices of the image. T, () (, ), or '(, ). T, () (, ), or '(, 3 ). T, () ( 0, ), or '(, 5 ). 8. Plot the points ', ', and. onnect the points to form '''. O O opright b Pearson Education, Inc. or its affiliates. ll Rights Reserved. hapter 9 8

4 Problem Writing a Rule to Describe a Translation Got It? The translation image of LMN is L'M'N' with L'(, ), M '(3, ), and N'(6, ). What is a rule that describes the translation? 9. hoose a pair of corresponding vertices. L( 6, ) L'(, ) 0. Find the horizontal change and the vertical change. ( 6 ) L M O N 3 ( ). The translation maps (, ) to ( 7, ). The translation rule is T 7, (LMN). Lesson heck Do ou UNDERSTND? Error nalsis Your friend sas the transformation PQR is a translation. Eplain and correct her error. Q. Identif the corresponding vertices in the statement PQR. and P and Q and R R P opright b Pearson Education, Inc. or its affiliates. ll Rights Reserved. 3. Identif the corresponding vertices from the diagram. and R and Q and. What was our friend s error? Eplain. Sample: The transformation statement PQR maps to P and to R so it is not a translation. 5. Write the correct transformation statement. Math Success heck off the vocabular words that ou understand. rigid motion preimage image translation Rate how well ou can use the properties of reflections. Need to review RQP Now I get it! P 9 Lesson 9-

5 9- Reflections Vocabular Review. is congruent to RST. Name the side that corresponds to. R ST S T r. In the diagram, r is the perpendicular bisector of. Vocabular uilder reflection (noun) ri flek shuh n Related Words: transformation, mirror image, flip, pre-image, image Main Idea: reflection is a transformation which flips a figure over a line of reflection resulting in a mirror image of the original figure. The orientation of the figure reverses. Eample: Figure D was reflected over the -ais to form figure LMNO. Figure LMNO is a reflection of figure D. Use Your Vocabular 3. Tell whether the pair of figures shows a reflection. Write es or no.. Reflect over the line. Draw and label its reflection. no D M L N O O es opright b Pearson Education, Inc. or its affiliates. ll Rights Reserved. hapter 9 30

6 Ke oncept Reflection cross a Line Reflection across a line m, called the line of reflection, is a transformation with these two properties: If a point is on line m, then the image of is itself (that is, ' ). is not on line m, then m is the perpendicular bisector of '. m that takes P to P' as R m (P) P'. m The preimage and its image are equidistant from the line of reflection. 5. If R m () ', and R m () ', then ''. 6. If R m () ''', then m m ''' 7. If 3 units, '' 3 units. 8. If m 55, m ''' 55 Problem Reflecting a Point cross a Line Point P has coordinates (3, ). Got It? R (P) P'. What are the coordinates of the image P'? opright b Pearson Education, Inc. or its affiliates. ll Rights Reserved. 9. Graph point P and the line of reflection. 0. Describe point P in relation to the line of reflection. O P = (3, ) Point P is units to the right/left of the line of reflection.. Describe image P' in relation to the line of reflection. Point P and image P' are the same distance from the line of reflection but in opposite directions, so image P' is reflection. units to the right/left of the line of. Graph image P'. Write its coordinates. 3 Lesson 9-

7 (, ) P O = P (3, ) left along the line through P that is perpendicular to the line of reflection. P and P' to the line of reflection are the same. The coordinates of P' are (, ). Problem Graphing a Reflection Image Got It? Graph, where (3, ), (0, ), and (, ). Graph and label R - ais ('''). 3. Graph. -ais as the dashed line of reflection.. ''' reflection that passes through point. ' so that the -ais is the perpendicular bisector of '. O and. ', ', and ' to form ''' Problem Using Properties of Reflections In the diagram, R t (G) G, R t (H) J, and R t (D) D. Got It? an ou use properties of reflections to prove that GHJ is equilateral? Eplain how ou know. 5. ll three sides are the same length. 6. Use the properties of reflections to compare the side lengths. H G D t J opright b Pearson Education, Inc. or its affiliates. ll Rights Reserved. GH GJ and HD JD hapter 9 3

8 For Eercises 7 and 8, circle the correct answer. 7. an ou determine if HJ GH GJ using the properties of reflection? es / no 8. The properties of reflections can/cannot be used to prove that GHJ is equilateral. Lesson heck Do ou UNDERSTND? Error nalsis classmate sketched R r () ' as shown in the diagram. a. Eplain our classmate s error. b. op point and line r and show the correct location of '. r 9. Use the properties of reflections to describe how the line of reflection r is related to '. Line r is the perpendicular bisector of '. 0. Eamine the classmate's drawing. What was the classmate's error? Eplain. nswers ma var. Sample: In the classmate's drawing, line r is not perpendicular to '.. Draw the correct location of '. opright b Pearson Education, Inc. or its affiliates. ll Rights Reserved. Math Success Need to review r heck off the vocabular words that ou understand. transformation reflection line of reflection Rate how well ou can use the properties of reflections Now I get it! 33 Lesson 9-

9 9-3 Rotations Vocabular Review. What is the name of the tool shown to the right? protractor nswers ma var. Sample: right angle. Draw a line segment to classif each angle measure. Then draw and label the angles on the tool acute angle right angle obtuse angle obtuse angle inches K acute angle Vocabular uilder rotation (noun) roh te shuh n Related Words: transformation, preimage, image, center of rotation, angle of rotation Main Idea: rotation is a transformation which turns a figure about a fied point called the center of rotation. Eample: propeller is fied to a boat or airplane at a center point. The blades rotate about that center of rotation. To map one blade of this propeller onto the net blade, rotate 90. Use Your Vocabular 3. Tell if each pair of figures shows a rotation. Write es or no. es no es opright b Pearson Education, Inc. or its affiliates. ll Rights Reserved. hapter 9 3

10 Rotations preserve distance, angle, and orientation of figures. Ke oncept Rotation bout a Point rotation of about a point Q, called the W center of rotation, is a transformation with these two properties: The image of Q is itself (that is, Q' Q). For an other point V, QV' QV and mvqv'. Q The positive number of degrees a figure rotates is Q V V W U The preimage V and its image Vare equidistant from the center of rotation. the angle of rotation. rotation about a point is a rigid motion. You write the rotation of UVW about point Q as r (, Q) (UVW) U'V'W'. U Problem Drawing a Rotation Image Got It? What is the image of LO for a 50 rotation about? O opright b Pearson Education, Inc. or its affiliates. ll Rights Reserved.. Fill in the blanks to develop our plan. The image of point is itself. So, '. The image of LO is L'O'. I need to rotate LO 50 around point. I will begin b drawing side L'. Then I will draw side O'. 5. ircle the tools ou will need to draw L'O'. L O L 5050 O L ruler compass protractor calculator 6. Write T for true and F for false net to each statement. F T T To draw L'O', I will rotate the preimage clockwise. The sides of the image must be congruent to the sides of the preimage. I need to use the compass and protractor to draw onl sides of the image. F Each angle in L'O' is 50 greater than each angle in LO. 35 Lesson 9-3

11 When a figure is rotated 90, 80, or 70 about the origin O in a coordinate plane, ou can use the following rules. Ke oncept Rotation in the oordinate Plane r (908, O) (, ) 5 (, ) G9(3, ) 6 G (, 3) O 6 r (808, O) (, ) 5 (, ) 6 G9(,3) 808 G(, 3) O 6 r (708, O) (, ) 5 (, ) G(, 3) O 6 G9(3, ) r (3608, O) (, ) 5 (, ) 6 G(, 3) O Problem Drawing Rotations in a oordinate Plane Got It? Graph r (70, O) (FGHI). 7. ircle the correct ordered pair ou use to find r (70, O) (, ). (, ) (, ) (, ) (, ) 8. Use the rule ou circled in Eample 7. Fill in the blanks to find the coordinates for each verte of the image and graph. F': r (70, O) (3, ) (, 3 ) G' : r (70, O) ( 3, ) (, 3 ) H' : r (70, O) (, ) (, ) I' : r (70, O) ( 0, ) (, 0 ) Problem 3 Using Properties of Rotations Got It? In the diagram, WXYZ is a parallelogram, and T is the midpoint of the diagonals. an ou use the properties of rotations to prove that WXYZ is a rhombus? Eplain. 9. Fill in the blanks or circle the word to complete each sentence. rhombus is a parallelogram / rectangle with two / four congruent sides. To prove that WXYZ is a rhombus I need to show that WX XY YZ ZW. Since the lengths of WX, XY, YZ, and ZW are known / unknown, ou can / cannot use a rotation to prove that WXYZ is a rhombus. X F G W T G9 3 I H9 I9 O 3 H 3 Y Z F9 opright b Pearson Education, Inc. or its affiliates. ll Rights Reserved. hapter 9 36

12 Lesson heck Do ou UNDERSTND? Reasoning Point P(, ) is rotated about the origin b 35 and then b 5. What are the coordinates of the image of point P? Eplain. 0. Find the sum of the angles of rotation. Fill in the blank to complete the equation Match each rotation with the rule ou can use to find the coordinates of the image. r (90, O) (, ) (, ) r (80, O) (, ) (, ) r (70, O) (, ) (, ) r (360, O) (, ) 80 (, ). Write the rule ou need to use to find the coordinates of point P'. r (80, O) (, ) (, ) 3. Use the graph to show point P and its image, point P'. 3 3 P (, ) O 3 nswers ma var. Sample: P9(, ) 3 opright b Pearson Education, Inc. or its affiliates. ll Rights Reserved.. Eplain how ou found the coordinates of the image of point P. Point P was rotated a total of 80. I used the rule r (80, O) (, ) (, ), so the image of P(, ) is P'(, ). Math Success heck off the vocabular words that ou understand. rotation point of rotation angle of rotation Rate how well ou can use the properties and rules of rotations. Need to review Now I get it! nswers ma var. Sample: 37 Lesson 9-3

13 9-6 Dilations Vocabular Review. Identif four isometries on the lines below. T ranslation, R otation, R eflection, G lide reflection. ircle the correct answer to complete the sentence. mapping that results in a change in the position, size, or shape of a geometric figure is called a/n isometr / congruence / transformation. Vocabular uilder SKL ˉ FK t r dı- Lˉ tion e Scale factor of a dilation (noun) Definition: The scale factor of a dilation is the ratio of a length of the preimage to the corresponding length in the image, with the image length alwas in the numerator. Eample: If the scale factor of a dilation is greater than one, the dilation is an enlargement. If the scale factor is less than one, the dilation is a reduction. Use Your Vocabular 3. Which of the following shows a dilation?. When a figure is transformed b a congruence transformation, what is true about the corresponding angles and the corresponding sides of the image and preimage? The corresponding angles are congruent and the corresponding sides are congruent. hapter 9 6 opright b Pearson Education, Inc. or its affiliates. ll Rights Reserved. Related Words: dilation, center of a dilation, reduction, enlargement

14 Ke oncept Dilation dilation with center of dilation and scale factor n, n 0, can be written as D (n, ). dilation is a transformation with the following properties: The image of is itself (that is, ' ). For an other point R, R' is on and R' n R, or n R' R'. Dilations preserve angle measure. P Q R Rn R P R Q 5. ircle the equation that is true for the dilation shown to the right. n X X' n X' X n Y Y' n Y' X X Y X Y Z Z Problem Finding a Scale Factor Got It? Is D(n, O)(JKLM) J'K'L'M' an enlargement or a reduction? What is the scale factor of the dilation? opright b Pearson Education, Inc. or its affiliates. ll Rights Reserved. Underline the correct choice to complete the sentence. 6. The image J'K'L'M' is larger / smaller than the preimage JKLM. 7. The image J'K'L'M' is a(n) enlargement / reduction. 8. Fill in the blanks to identif the coordinates of the preimage JKLM. J( 0, ), K( 6, 0 ), L( 6, ), M(, ), 9. Fill in the blanks to identif the coordinates of the image J'K'L'M'. J'( 0, ), K'( 3, 0 ), L'( 3, ), M'(, ), J K O 0. Use the distance formula d ( ) ( ) to find the lengths of the corresponding sides of the preimage JK and the image J'K' in simplest radical form. Fill in the blanks. JK ( 6 0 ) ( 0 ) 0 J'K' ( 3 0 ) ( 0 ) 0. Fill in the blanks to complete the sentence. The scale factor of the dilation is n J'K' JK 0 0 M M 3 J L K L 7 Lesson 9-6

15 Problem Finding a Dilation Image Got It? What are the coordinates of the vertices of D (PZG)? Fill in the blanks to complete the sentence.. The center of the dilation is the origin and the scale factor is. Underline the correct word to complete the sentence. 3 3 O 3 3. ecause the scale factor is less than, the dilation is a(n) enlargement / reduction.. Use D (, ), to find the coordinates of each image verte and plot P'Z'G'. 3 D (P), 0 ; or P', 0 D (Z) 3, ; or Z' 3, D (G) 0, ; or G' 0, Problem 3 Using a Scale Factor to Find a Length Got It? The height of a document on our computer screen is 0. cm. When ou change the zoom setting on our screen from 00% to 5%, the new image of our document is a dilation of the previous image with scale factor 0.5. What is the height of the new image? Underline the correct word to complete the sentence. 5. ecause the scale factor is 0.5, the dilation is a(n) enlargement/reduction. Fill in the blanks to complete the statements. 6. scale factor of 0.5 tells ou that the ratio of the image length to the actual length is 0.5. image length actual length So, image length scale factor actual length cm opright b Pearson Education, Inc. or its affiliates. ll Rights Reserved. hapter 9 8

16 Lesson heck Do ou UNDERSTND? Error nalsis The gra figure is a dilation image of the black figure for a dilation with center. student made an error when asked to find the scale factor. Eplain and correct the error. 6 3 n= 6 = 3 Use the diagram to complete Eercises 7 to. 7. ircle the image. '' 8. ircle the preimage. '' Draw line segments to identif three pairs of corresponding sides of the image and the preimage. ' '' ' Fill in the blanks to complete the statements. 0. Find the side lengths. ' ' 3 opright b Pearson Education, Inc. or its affiliates. ll Rights Reserved. 8 Math Success. Find the scale factor for two pairs of corresponding sides. n ' ' n 8. Eplain and correct the error on the line below. nswers ma var. Sample: The student found the ratio of ' to '. The scale factor n of a dilation is the ratio of a length of the preimage to the corresponding length in the image, with the image length alwas in the numerator. In this case, the scale factor is the ratio of ' to, or to. heck off the vocabular words that ou understand. dilation center of dilation scale factor of a dilation enlargement Rate how well ou can use dilations. reduction Need to review Now I get it! 9 Lesson 9-6