How can you enlarge or reduce a figure in the coordinate plane? Dilate. ACTIVITY: Comparing Triangles in a Coordinate Plane.

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1 . Dilations How can ou enlarge or reduce a figure in the coordinate plane? Dilate When ou have our ees checked, the optometrist sometimes dilates one or both of the pupils of our ees. ACTIVITY: Comparing Triangles in a Coordinate Plane Work with a partner. Write the coordinates of the vertices of the blue triangle. Then write the coordinates of the vertices of the red triangle. a. How are the two sets of coordinates related? b. How are the two triangles related? Eplain our reasoning. Ź Ź Ź Ź Ź Ź Geometr In this lesson, ou will identif dilations. dilate figures in the coordinate plane. use more than one transformation to find images of figures. Ź Ź Ź Ź c. Draw a green triangle whose coordinates are twice the values of the corresponding coordinates of the blue triangle. How are the green and blue triangles related? Eplain our reasoning. d. How are the coordinates of the red and green triangles related? How are the two triangles related? Eplain our reasoning. 8 Chapter ms_blue pe_00.indd 8 Transformations // :9:0 PM

cm 8 9 0 ACTIVITY: Drawing Triangles in a Coordinate Plane Work with a partner. a. Draw the triangle whose vertices are (0, ), (, ), and (, ). b.

Math Practice Use Prior Results What are the four tpes of transformations ou studied in this chapter? What information can ou use to fill in our table? c. Repeat part (b) b multipling b instead of.

2 cm ACTIVITY: Drawing Triangles in a Coordinate Plane Work with a partner. a. Draw the triangle whose vertices are (0, ), (, ), and (, ). b. Multipl each coordinate of the vertices b to obtain three new vertices. Draw the triangle given b the three new vertices. How are the two triangles related? Math Practice Use Prior Results What are the four tpes of transformations ou studied in this chapter? What information can ou use to fill in our table? c. Repeat part (b) b multipling b instead of. ACTIVITY: Summarizing Transformations Work with a partner. Make a table that summarizes the relationships between the original figure and its image for the four tpes of transformations ou studied in this chapter.. IN YUR WN WRDS How can ou enlarge or reduce a figure in the coordinate plane?. Describe how knowing how to enlarge or reduce figures in a technical drawing is important in a career such as drafting. Use what ou learned about dilations to complete Eercises on page 8. Section. Dilations 8

. Lesson Lesson Tutorials A dilation is a transformation in which a figure is made larger or smaller with respect to a point called the center of dilation.

3 . Lesson Lesson Tutorials A dilation is a transformation in which a figure is made larger or smaller with respect to a point called the center of dilation. Center of dilation EXAMPLE Ke Vocabular dilation, p. 8 center of dilation, p. 8 scale factor, p. 8 Identifing a Dilation Tell whether the blue figure is a dilation of the red figure. a. b. Lines connecting corresponding vertices meet at a point. The figures have the same size and shape. The red figure slides to form the blue figure. So, the blue figure is a dilation of the red figure. So, the blue figure is not a dilation of the red figure. It is a translation. Eercises Tell whether the blue figure is a dilation of the red figure. Eplain... In a dilation, the original figure and its image are similar. The ratio of the side lengths of the image to the corresponding side lengths of the original figure is the scale factor of the dilation. Dilations in the Coordinate Plane Words To dilate a figure with respect to the origin, multipl the coordinates of each verte b the scale factor k. Algebra (, ) (k, k) A B C B C When k >, the dilation is an enlargement. When k > 0 and k <, the dilation is a reduction. 8 Chapter Transformations

Stud Tip EXAMPLE Multipl each - and -coordinate b the scale factor. You can check our answer b drawing a line from the origin through each verte of the original figure.

ABC (, ) A B C A(, ) (, ) A (, 9) B(, ) (, ) B (, 9) C(, ) (, ) C (, ) The image is shown at the right. The dilation is an enlargement because the scale factor is greater than.

4 Stud Tip EXAMPLE Multipl each - and -coordinate b the scale factor. You can check our answer b drawing a line from the origin through each verte of the original figure. The vertices of the image should lie on these lines. Dilating a Figure Draw the image of Triangle ABC after a dilation with a scale factor of. Identif the tpe of dilation. ABC (, ) A B C A(, ) (, ) A (, 9) B(, ) (, ) B (, 9) C(, ) (, ) C (, ) The image is shown at the right. The dilation is an enlargement because the scale factor is greater than. 9 8 A B C B C EXAMPLE Multipl each - and -coordinate b the scale factor 0.. Dilating a Figure Draw the image of Rectangle WXYZ after a dilation with a scale factor of 0.. Identif the tpe of dilation. WXYZ (0., 0.) W X Y Z W(, ) (0. ( ), 0. ( )) W (, ) X(, 8) (0. ( ), 0. 8) X (, ) Y (, 8) (0., 0. 8) Y (, ) Z (, ) (0., 0. ( )) Z (, ) X X 0 Y Y The image is shown at the right. The dilation is a reduction because the scale factor is greater than 0 and less than. W W 8 Z Z Eercises 8. WHAT IF? Triangle ABC in Eample is dilated b a scale factor of. What are the coordinates of the image?. WHAT IF? Rectangle WXYZ in Eample is dilated b a scale factor of. What are the coordinates of the image? Section. Dilations 8

5 EXAMPLE Using More than ne Transformation The vertices of a trapezoid are A(, ), B(, ), C(0, ), and D(0, ). Dilate the trapezoid with respect to the origin using a scale factor of. Then translate it units right and units up. What are the coordinates of the image? Draw ABCD. Then dilate it with respect to the origin using a scale factor of. B A B C C D D B C D Translate the dilated figure units right and units up. The coordinates of the image are A (, 0), B (, ), C (, ), and D (, 0). The image of a translation, reflection, or rotation is congruent to the original figure, and the image of a dilation is similar to the original figure. So, two figures are similar when one can be obtained from the other b a sequence of translations, reflections, rotations, and dilations. EXAMPLE Describing a Sequence of Transformations The red figure is similar to the blue figure. Describe a sequence of transformations in which the blue figure is the image of the red figure. From the graph, ou can see that the blue figure is one-half the size of the red figure. So, begin with a dilation with respect to the origin using a scale factor of. After dilating, ou need to flip the figure in the -ais. So, one possible sequence of transformations is a dilation with respect to the origin using a scale factor of followed b a reflection in the -ais. Eercises 8. In Eample, use a scale factor of in the dilation. Then rotate the figure 80 about the image of verte C. What are the coordinates of the image?. In Eample, can ou reflect the red figure first, and then perform the dilation to obtain the blue figure? Eplain. 8 Chapter Transformations

6 . Eercises Help with Homework. VCABULARY How is a dilation different from other transformations?. VCABULARY For what values of scale factor k is a dilation called an enlargement? a reduction?. REASNING Which figure is not a dilation of the blue figure? Eplain. 9+(-)= +(-)= +(-9)= 9+(-)= Draw the triangle with the given vertices. Multipl each coordinate of the vertices b, and then draw the new triangle. How are the two triangles related?. (0, ), (, ), (, 0). (, ), (, ), (, ). (, ), (, ), (, ) Tell whether the blue figure is a dilation of the red figure The vertices of a figure are given. Draw the figure and its image after a dilation with the given scale factor. Identif the tpe of dilation.. A(, ), B(, ), C(, ); k =. D(0, ), E(, ), F(, ); k = 0.. G(, ), H(, ), J(, ); k = 0.. M(, ), N(, ), P(, ); k =. Q(, 0), R(, ), T(, ), U(, 0); k = 8. V(, ), W(, ), X(, ), Y(, ); k = Section. Dilations 8

J 8 J B A C C B 8 Y X Y Z Z 0 M M 8 0 K K L L The vertices of a figure are given. Find the coordinates of the figure after the transformations given.. A(, ), B(, ), C(, ), D(, ) Reflect in the -ais.

7 9. ERRR ANALYSIS Describe and correct the error in listing the coordinates of the image after a dilation with a scale factor of. Vertices of ABC (, ) A B C A(, ) (, ) A (, 0) B(, 0) (, 0) B (, 0) C(, 0) (, 0) C (8, 0) The blue figure is a dilation of the red figure. Identif the tpe of dilation and find the scale factor. 0.. X. J 8 J B A C C B 8 Y X Y Z Z 0 M M 8 0 K K L L The vertices of a figure are given. Find the coordinates of the figure after the transformations given.. A(, ), B(, ), C(, ), D(, ) Reflect in the -ais. Then dilate with respect to the origin using a scale factor of.. F( 9, 9), G(, ), H(, 9) Dilate with respect to the origin using a scale factor of. Then translate units up.. J(, ), K(, ), L(, ) Rotate 90 clockwise about the origin. Then dilate with respect to the origin using a scale factor of.. P(, ), Q(, ), R(, ), S(, ) Dilate with respect to the origin using a scale factor of. Then dilate with respect to the origin using a scale factor of 0.. The red figure is similar to the blue figure. Describe a sequence of transformations in which the blue figure is the image of the red figure STRUCTURE In Eercises and 8, is the blue figure still the image of the red figure when ou perform the sequence in the opposite order? Eplain. 88 Chapter Transformations

8 0. PEN-ENDED Draw a rectangle on a coordinate plane. Choose a scale factor of,,, or, and then dilate the rectangle. How man times greater is the area of the image than the area of the original rectangle?. SHADW PUPPET You can use a flashlight and a shadow puppet (our hands) to project shadows on the wall. a. Identif the tpe of dilation. b. What does the flashlight represent? c. The length of the ears on the shadow puppet is inches. The length of the ears on the shadow is inches. What is the scale factor? d. Describe what happens as the shadow puppet moves closer to the flashlight. How does this affect the scale factor?. REASNING A triangle is dilated using a scale factor of. The image is then dilated using a scale factor of. What scale factor could ou use to dilate the original triangle to get the final image? Eplain. CRITICAL THINKING The coordinate notation shows how the coordinates of a figure are related to the coordinates of its image after transformations. What are the transformations? Are the figure and its image similar or congruent? Eplain.. (, ) ( +, ). (, ) (, ) (. (, ), ). STRUCTURE How are the transformations ( +, ) and (( + ), ( + )) different?. Problem The vertices of a trapezoid are A(, ), B(, ), C(, ), and Solving D(, ). Dilate the trapezoid with respect to verte A using a scale factor of. What are the coordinates of the image? Eplain the method ou used. Tell whether the angles are complementar or supplementar. Then find the value of. (Skills Review Handbook) í í í ( à 0)í í ( Ź 0)í. MULTIPLE CHICE Which quadrilateral is not a parallelogram? (Skills Review Handbook) A rhombus B trapezoid C square D rectangle Section. ms_blue pe_00.indd 89 Dilations 89 // :9:9 PM

Rotate. A bicycle wheel can rotate clockwise or counterclockwise. ACTIVITY: Three Basic Ways to Move Things

Rotate. A bicycle wheel can rotate clockwise or counterclockwise. ACTIVITY: Three Basic Ways to Move Things . Rotations object in a plane? What are the three basic was to move an Rotate A biccle wheel can rotate clockwise or counterclockwise. 0 0 0 9 9 9 8 8 8 7 6 7 6 7 6 ACTIVITY: Three Basic Was to Move Things