Properties of Rotations 8.10.A. Sketch the image of the rotation. Label the images of points A, B, and C as A, B, and C.

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1 ? LESSN 1.3 ESSENTIL QUESTIN Properties of Rotations How do ou describe the properties of orientation and congruence of rotations? Two-dimensional shapes Generalize the properties of orientation and congruence of rotations of twodimensional shapes on a coordinate plane. EXPLRE TIVITY Eploring Rotations rotation is a transformation that turns a figure around a given point called the center of rotation. The image has the same size and shape as the preimage. The triangle shown on the grid is the preimage. You will use the origin as the center of rotation. Houghton Mifflin Harcourt Publishing ompan Image redits: IK/Fotolia E Trace triangle onto a piece of paper. ut out our traced triangle. Rotate our triangle 90 counterclockwise about the origin. The side of the triangle that lies along the -ais should now lie along the -ais. Sketch the image of the rotation. Label the images of points,, and as,, and. escribe the motion modeled b the rotation. Rotate about the origin. degrees heck that the motion ou described in is the same motion that maps point onto, point onto, and point onto. Reflect 1. ommunicate Mathematical Ideas How are the size and the orientation of the triangle affected b the rotation?. Rotate triangle 90 clockwise about the origin. Sketch the result on the coordinate grid above. Label the image vertices,, and. Lesson 1.3 3

2 EXPLRE TIVITY Properties of Rotations Use trapezoid TRP to investigate the properties of rotations. Trace the trapezoid onto a piece of paper. Include the portion of the - and -aes bordering the third quadrant. ut out our tracing. Place our trapezoid and aes on top of those in the figure. Then use the aes to help rotate our trapezoid 180 counterclockwise about the origin. Sketch the image of the rotation of our trapezoid in this new location. Label the vertices of the image T, R,, and P. - R Use a ruler to measure the sides of trapezoid TRP in centimeters. - T P TR = R = P = TP = Use a ruler to measure the sides of trapezoid T R P in centimeters. T R = R = P = T P = E What do ou notice about the lengths of corresponding sides of the two figures? F G H I Use a protractor to measure the angles of trapezoid TRP. m T = m R = m = m P = Use a protractor to measure the angles of trapezoid T R P. m T = m R = m = m P = What do ou notice about the measures of corresponding angles of the two figures? Which sides of trapezoid TRP are parallel? Which sides of trapezoid T R P are parallel? What do ou notice? Houghton Mifflin Harcourt Publishing ompan 3 Unit

3 Reflect 3. Make a onjecture Use our results from E, H, and I to make a conjecture about rotations.. Place our tracing back in its original position. Then perform a 180 clockwise rotation about the origin. ompare the result. Graphing Rotations To rotate a figure in the coordinate plane, rotate each of its vertices. Then connect the vertices to form the image. EXMPLE Math n the Spot m.hrw.com The figure shows triangle. Graph the image of triangle after a rotation of 90 clockwise. STEP 1 Rotate the figure clockwise from the -ais to the -ais. Point will still be at (0, 0). - nimated Math m.hrw.com Houghton Mifflin Harcourt Publishing ompan STEP Point is units to the left of the -ais, so point is units above the -ais. Point is units to the right of the -ais, so point is units below the -ais. onnect,, and to form the image triangle. Reflect. Is the image congruent to the preimage? How do ou know? - - Math Talk Mathematical Processes How is the orientation of the triangle affected b the rotation? - Lesson

4 Personal Math Trainer nline ssessment and Intervention YUR TURN Graph the image of quadrilateral after each rotation m.hrw.com clockwise 8. Find the coordinates of Point after a 90 counterclockwise rotation followed b a 180 rotation. Guided Practice 1. Vocabular rotation is a transformation that turns a figure around a given called the center of rotation. Siobhan rotates a right triangle 90 counterclockwise about the origin.. How does the orientation of the image of the triangle compare with the orientation of the preimage? (Eplore ctivit 1) 3. Is the image of the triangle congruent to the preimage? (Eplore ctivit )? raw the image of the figure after the given rotation about the origin. (Eample 1). 90 counterclockwise. 180 E F G - - ESSENTIL QUESTIN HEK-IN. What are the properties of rotations? Houghton Mifflin Harcourt Publishing ompan 38 Unit

5 ? LESSN 1. lgebraic Representations of Transformations ESSENTIL QUESTIN lgebraic Representations of Translations The rules shown in the table describe how coordinates change when a figure is translated up, down, right, and left on the coordinate plane. Translations Two-dimensional shapes Eplain the effect of translations, reflections over the - or -ais, and rotations limited to 90, 180, 70, and 30 as applied to twodimensional shapes on a coordinate plane using an algebraic representation. How can ou describe the effect of a translation, rotation, or reflection on coordinates using an algebraic representation? Math n the Spot m.hrw.com Right a units dd a to the -coordinate: (, ) ( + a, ) Left a units Subtract a from the -coordinate: (, ) ( - a, ) Up b units dd b to the -coordinate: (, ) (, + b) own b units Subtract b from the -coordinate: (, ) (, - b) EXMPLE Houghton Mifflin Harcourt Publishing ompan Triangle XYZ has vertices X(0, 0), Y(, 3), and Z(, -1). Find the vertices of triangle X Y Z after a translation of 3 units to the right and 1 unit down. Then graph the triangle and its image. dd 3 to the -coordinate of each verte and subtract 1 from the STEP 1 ppl the rule to find the vertices of the image. -coordinate of each verte. STEP Vertices of XYZ Rule: ( + 3, - 1) Vertices of X Y Z X(0, 0) (0 + 3, 0-1) X (3, -1) Y(, 3) ( + 3, 3-1) Y (, ) Z(, -1) ( + 3, -1-1) Z (7, -) Graph triangle XYZ and its image. Y 3 X -3 X Y Z 7 Z Math Talk Mathematical Processes When ou translate a figure to the left or right, which coordinate do ou change? Lesson 1. 31

6 YUR TURN Personal Math Trainer nline ssessment and Intervention m.hrw.com 1. rectangle has vertices at (0, -), (0, 3), (3, -), and (3, 3). What are the coordinates of the vertices of the image after the translation (, ) ( -, - 3)? escribe the translation. Math n the Spot m.hrw.com lgebraic Representations of Reflections The signs of the coordinates of a figure change when the figure is reflected across the -ais and -ais. The table shows the rules for changing the signs of the coordinates after a reflection. Reflections cross the -ais Multipl each -coordinate b -1: (, ) (, -) cross the -ais Multipl each -coordinate b -1: (, ) (-, ) M Notes EXMPLE Rectangle RSTU has vertices R(-, -1), S(-1, -1), T(-1, -3), and U(-, -3). Find the vertices of rectangle R S T U after a reflection across the -ais. Then graph the rectangle and its image. Multipl the -coordinate of STEP 1 ppl the rule to find the vertices of the image. each verte b -1. STEP Vertices of RSTU Rule: (-1, ) Vertices of R S T U R(-, -1) (-1 (-), -1) R (, -1) S(-1, -1) (-1 (-1), -1) S (1, -1) T(-1, -3) (-1 (-1), -3) T (1, -3) U(-, -3) (-1 (-), -3) U (, -3) Graph rectangle RSTU and its image. 3 Houghton Mifflin Harcourt Publishing ompan R S S R U T T U 3 Unit

7 YUR TURN. Triangle has vertices (-, ), (0, ), and (3, -1). Find the vertices of triangle after a reflection across the -ais. lgebraic Representations of Rotations When points are rotated about the origin, the coordinates of the image can be found using the rules shown in the table. 90 clockwise 90 counterclockwise Rotations Multipl each -coordinate b -1; then switch the - and -coordinates: (, ) (, -) Multipl each -coordinate b -1; then switch the - and -coordinates: (, ) (-, ) 180 Multipl both coordinates b -1: (, ) (-, -) Personal Math Trainer nline ssessment and Intervention m.hrw.com Math n the Spot m.hrw.com EXMPLE Quadrilateral has vertices at (-, ), (-3, ), (, 3), and (0, 0). Find the vertices of quadrilateral after a 90 clockwise rotation. Then graph the quadrilateral and its image. Multipl the -coordinate of each verte b -1, and then STEP 1 ppl the rule to find the vertices of the image. switch the - and -coordinates. Vertices of Rule: (, -) Vertices of (-, ) (, -1 (-)) (, ) Houghton Mifflin Harcourt Publishing ompan STEP (-3, ) (, -1 (-3)) (, 3) (, 3) (3, -1 ) (3, -) (0, 0) (0, -1 0) (0, 0) Graph the quadrilateral and its image. - - Math Talk Mathematical Processes Eplain how to use the 90 rotation rule to develop a rule for a 30 rotation. - - Lesson 1. 33

8 Reflect 3. ommunicate Mathematical Ideas How would ou find the vertices of an image if a figure were rotated 70 clockwise? Eplain. YUR TURN Personal Math Trainer nline ssessment and Intervention m.hrw.com. triangle has vertices at J(-, -), K(1, ), and L(, ). What are the coordinates of the vertices of the image after the triangle is rotated 90 counterclockwise? Guided Practice 1. Triangle XYZ has vertices X(-3, -), Y(-1, 0), and Z(1, -). Find the vertices of triangle X Y Z after a translation of units to the right. Then graph the triangle and its image. (Eample 1) -3 Y 3 7. escribe what happens to the - and -coordinates after a point is reflected across the -ais. (Eample ) X? 3. Use the rule (, ) (, -) to graph the image of the triangle at right. Then describe the transformation. (Eample 3) ESSENTIL QUESTIN HEK-IN. How do the - and -coordinates change when a figure is translated right a units and down b units? -7 Z -3 Houghton Mifflin Harcourt Publishing ompan 3 Unit

9 ? LESSN 13.1 ESSENTIL QUESTIN Properties of ilations How do ou describe the properties of dilations? Proportionalit 8.3. ompare and contrast the attributes of a shape and its dilation(s) on a coordinate plane. lso 8.3., 8.10., EXPLRE TIVITY 1 Eploring ilations The missions that placed 1 astronauts on the moon were controlled at the Johnson Space enter in Houston. The to models at the right are scaled-down replicas of the Saturn V rocket that powered the moon flights. Each replica is a transformation called a dilation. Unlike the other transformations ou have studied translations, rotations, and reflections dilations change the size (but not the shape) of a figure. 8.3., enter of dilation R' Ever dilation has a fied point called the center of dilation located where the lines connecting corresponding parts of figures intersect. Triangle R S T is a dilation of triangle RST. Point is the center of dilation. R S T S' T' Use a ruler to measure segments _ R, _ R, _ S, _ S, _ T, and _ T to the nearest millimeter. Record the measurements and ratios in the table. R R R R S S S S T T T T Houghton Mifflin Harcourt Publishing ompan Write a conjecture based on the ratios in the table. Measure and record the corresponding side lengths of the triangles. R S RS R S RS S T ST S T ST Write a conjecture based on the ratios in the table. R T RT R T RT E Measure the corresponding angles and describe our results. Lesson

10 EXPLRE TIVITY 1 (cont d) Reflect 1. re triangles RST and R S T similar? Wh or wh not?. ompare the orientation of a figure with the orientation of its dilation. EXPLRE TIVITY 8.3. Eploring ilations on a oordinate Plane In this activit ou will eplore how the coordinates of a figure on a coordinate plane are affected b a dilation. omplete the table. Record the - and -coordinates of the points in the two figures and the ratios of the -coordinates and the -coordinates. - ' ' ' ' - Verte Verte Ratio of -coordinates ( ) Ratio of -coordinates ( ) Houghton Mifflin Harcourt Publishing ompan Write a conjecture about the ratios of the coordinates of a dilation image to the coordinates of the original figure. 3 Unit

11 Reflect 3. In Eplore ctivit 1, triangle R S T was larger than triangle RST. How is the relationship between quadrilateral and quadrilateral different? Math Talk Mathematical Processes How are dilations different from the other transformations ou have learned about? Finding a Scale Factor s ou have seen in the two activities, a dilation can produce a larger figure (an enlargement) or a smaller figure (a reduction). The scale factor describes how much the figure is enlarged or reduced. The scale factor is the ratio of a length of the image to the corresponding length on the original figure. In Eplore ctivit 1, the side lengths of triangle R S T were twice the length of those of triangle RST, so the scale factor was. In Eplore ctivit, the side lengths of quadrilateral were half those of quadrilateral, so the scale factor was 0.. Math n the Spot m.hrw.com EXMPLE Houghton Mifflin Harcourt Publishing ompan n art suppl store sells several sizes of drawing triangles. ll are dilations of a single basic triangle. The basic triangle and one of its dilations are shown on the grid. Find the scale factor of the dilation. STEP 1 STEP Use the coordinates to find the lengths of the sides of each triangle. Triangle : = = 3 Triangle : = = Find the ratios of the corresponding sides. = _ = = _ 3 = The scale factor of the dilation is ' ' ' 8 10 Since the scale factor is the same for all corresponding sides, ou can record just two pairs of side lengths. Use one pair as a check on the other. Reflect. Is the dilation an enlargement or a reduction? How can ou tell? Lesson

12 YUR TURN Personal Math Trainer nline ssessment and Intervention. Find the scale factor of the dilation E m.hrw.com Math Talk Mathematical Processes Which scale factors lead to enlargements? Which scale factors lead to reductions? ' G G' E' F F' 8 10 Guided Practice Use triangles and for 1. (Eplore ctivities 1 and, Eample 1) 1. For each pair of corresponding vertices, find the ratio of the -coordinates and the ratio of the -coordinates. ratio of -coordinates = ratio of -coordinates =. I know that triangle is a dilation of triangle because the ratios of the corresponding ' ' -coordinates are and the ratios of the corresponding -coordinates are. ' 3. The ratio of the lengths of the corresponding sides of triangle and? triangle equals.. The corresponding angles of triangle and triangle are.. The scale factor of the dilation is. ESSENTIL QUESTIN HEK-IN. How can ou find the scale factor of a dilation? Houghton Mifflin Harcourt Publishing ompan 3 Unit

13 ? LESSN 13. lgebraic Representations of ilations ESSENTIL QUESTIN Proportionalit 8.3. Use an algebraic representation to eplain the effect of a given positive rational scale factor applied to two-dimensional figures on a coordinate plane with the origin as the center of dilation. lso 8.3., How can ou describe the effect of a dilation on coordinates using an algebraic representation? EXPLRE TIVITY Graphing Enlargements When a dilation in the coordinate plane has the origin as the center of dilation, ou can find points on the dilated image b multipling the - and -coordinates of the original figure b the scale factor. For scale factor k, the algebraic representation of the dilation is (, ) (k, k). For enlargements, k > 1. The figure shown on the grid is the preimage. The center of dilation is the origin. List the coordinates of the vertices of the preimage in the first column of the table. Preimage (, ) Image (3, 3) 7 (, ) (, ) Houghton Mifflin Harcourt Publishing ompan What is the scale factor for the dilation? ppl the dilation to the preimage and write the coordinates of the vertices of the image in the second column of the table. Sketch the image after the dilation on the coordinate grid Math Talk Mathematical Processes What effect would the dilation (, ) (, ) have on the radius of a circle? Lesson

14 EXPLRE TIVITY 1 (cont d) Reflect 1. How does the dilation affect the length of line segments?. How does the dilation affect angle measures? EXPLRE TIVITY 8.3. Graphing Reductions For scale factors between 0 and 1, the image is smaller than the preimage. This is called a reduction. The arrow shown is the preimage. The center of dilation is the origin. List the coordinates of the vertices of the preimage in the first column of the table. Preimage (, ) Image 1_ 1_ (, ) What is the scale factor for the dilation? - ppl the dilation to the preimage and write the coordinates of the vertices of the image in the second column of the table. Sketch the image after the dilation on the coordinate grid. Reflect 3. How does the dilation affect the length of line segments? Houghton Mifflin Harcourt Publishing ompan. How would a dilation with scale factor 1 affect the preimage? 370 Unit

15 enter of ilation utside the Image The center of dilation can be inside or outside the original image and the dilated image. The center of dilation can be anwhere on the coordinate plane as long as the lines that connect each pair of corresponding vertices between the original and dilated image intersect at the center of dilation. Math n the Spot m.hrw.com EXMPLE Graph the image of after a dilation with the origin as its center and a scale factor of 3. What are the vertices of the image? STEP 1 Multipl each coordinate of the vertices of b 3 to find the vertices of the dilated image. 8 (, ) (3, 3) (1, 1) (1 3, 1 3) (3, 3) (3, 1) (3 3, 1 3) (9, 3) (1, 3) (1 3, 3 3) (3, 9) The vertices of the dilated image are (3, 3), (9, 3), and (3, 9). 8 STEP Graph the dilated image. 8 ' Math Talk Mathematical Processes Houghton Mifflin Harcourt Publishing ompan YUR TURN ' 8. Graph the image of XYZ after a dilation with a scale factor of 1_ and the origin as 3 its center. Then write an algebraic rule to describe the dilation. ' 8 escribe how ou can check graphicall that ou have drawn the image triangle correctl. Z X 8 Y Personal Math Trainer nline ssessment and Intervention m.hrw.com Lesson

16 Guided Practice 1. The grid shows a diamond-shaped preimage. Write the coordinates of the vertices of the preimage in the first column of the table. Then appl the dilation (, ) ( 3 _, 3 _ ) and write the coordinates of the vertices of the image in the second column. Sketch the image of the figure after the dilation. (Eplore ctivities 1 and ) Preimage Image (, 0) (3, 0) Graph the image of each figure after a dilation with the origin as its center and the given scale factor. Then write an algebraic rule to describe the dilation. (Eample 1). scale factor of scale factor of 1_ I H F G? 8 ESSENTIL QUESTIN HEK-IN 8. dilation of (, ) (k, k) when 0 < k < 1 has what effect on the figure? What is the effect on the figure when k > 1? Houghton Mifflin Harcourt Publishing ompan 37 Unit