Dilations. Essential Question What does it mean to dilate a figure?

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1 OMMON OE Learning Standards HSG-O.. HSG-ST..a HSG-ST..b LOOKING FO STUTUE.5 To be proficient in math, ou need to look closel to discern a pattern or structure. Dilations Essential Question What does it mean to dilate a figure? Dilating a Triangle in a oordinate lane Work with a partner. Use dnamic geometr software to draw an triangle and label it B. a. Dilate B using a scale factor of and a center of dilation at the origin to form B. ompare the coordinates, side lengths, and angle measures of B and B. Sample B 6 oints 5 (, ) B B(, ) (, ) Segments B =. B =. =. ngles 0 D m = 7.57 m B = 6.87 m = 7.57 b. epeat part (a) using a scale factor of. c. What do the results of parts (a) and (b) suggest about the coordinates, side lengths, and angle measures of the image of B after a dilation with a scale factor of k? Dilating Lines in a oordinate lane Work with a partner. Use dnamic geometr software to draw B that passes through the origin and that does not pass through the origin. a. Dilate B using a scale factor of and a center of dilation at the origin. Describe the image. b. Dilate using a scale factor of and a center of dilation at 0 the origin. Describe the image. B c. epeat parts (a) and (b) using a scale factor of. d. What do ou notice about dilations of lines passing through the center of dilation and dilations of lines not passing through the center of dilation? ommunicate Your nswer. What does it mean to dilate a figure? Sample. epeat Eploration using a center of dilation at a point other than the origin. oints (, ) B(0, 0) (, 0) 0 Lines + = 0 + = Section.5 Dilations 07

2 .5 Lesson What You Will Learn ore Vocabular dilation, p. 08 center of dilation, p. 08 scale factor, p. 08 enlargement, p. 08 reduction, p. 08 Identif and perform dilations. Solve real-life problems involving scale factors and dilations. Identifing and erforming Dilations ore oncept Dilations dilation is a transformation in which a figure is enlarged or reduced with respect to a fied point called the center of dilation and a scale factor k, which is the ratio of the lengths of the corresponding sides of the image and the preimage. dilation with center of dilation and scale factor k maps ever point in a figure to a point so that the following are true. If is the center point, then =. If is not the center point, then the image Q point lies on. The scale factor k is a positive number such that k =. Q ngle measures are preserved. dilation does not change an line that passes through the center of dilation. dilation maps a line that does not pass through the center of dilation to a parallel line. In the figure above,, Q Q, and Q Q. When the scale factor k >, a dilation is an enlargement. When 0 < k <, a dilation is a reduction. Identifing Dilations EDING The scale factor of a dilation can be written as a fraction, decimal, or percent. Find the scale factor of the dilation. Then tell whether the dilation is a reduction or an enlargement. a. b a. Because = 8, the scale factor is k =. So, the dilation is an enlargement. b. Because = 8 0, the scale factor is k =. So, the dilation is a reduction. 5 Monitoring rogress Help in English and Spanish at BigIdeasMath.com. In a dilation, = and =. Find the scale factor. Then tell whether the dilation is a reduction or an enlargement. 08 hapter Transformations

3 EDING DIGMS In this chapter, for all of the dilations in the coordinate plane, the center of dilation is the origin unless otherwise noted. ore oncept oordinate ule for Dilations If (, ) is the preimage of a point, then its image after a dilation centered at the origin (0, 0) with scale factor k is the point (k, k). Dilating a Figure in the oordinate lane (, ) (k, k) (, ) (k, k) Graph B with vertices (, ), B(, ), and (, ) and its image after a dilation with a scale factor of. Use the coordinate rule for a dilation with k = to find the coordinates of the vertices of the image. Then graph B and its image. (, ) (, ) (, ) (, ) B(, ) B (8, ) (, ) (8, ) B 6 B Notice the relationships between the lengths and slopes of the sides of the triangles in Eample. Each side length of B is longer than its corresponding side b the scale factor. The corresponding sides are parallel because their slopes are the same. Dilating a Figure in the oordinate lane Graph quadrilateral KLMN with vertices K(, 6), L(0, 6), M(, ), and N(, ) and its image after a dilation with a scale factor of. Use the coordinate rule for a dilation with k = to find the coordinates of the vertices of the image. Then graph quadrilateral KLMN and its image. (, ) (, ) K(, 6) K (, ) L(0, 6) L (0, ) M(, ) M (, ) N(, ) N (, ) K N L K L M N M Monitoring rogress Help in English and Spanish at BigIdeasMath.com Graph Q and its image after a dilation with scale factor k.. (, ), Q(, 0), (0, ); k =. (5, 5), Q(0, 5), (0, 5); k = 0. Section.5 Dilations 09

4 onstructing a Dilation Use a compass and straightedge to construct a dilation of Q with a scale factor of. Use a point outside the triangle as the center of dilation. Step Step Step Q Q Q Q Q Draw a triangle Draw Q and choose the center of the dilation outside the triangle. Draw ras from through the vertices of the triangle. Use a compass Use a compass to locate on so that = (). Locate Q and using the same method. onnect points onnect points, Q, and to form Q. scale factor k center of dilation preimage In the coordinate plane, ou can have scale factors that are negative numbers. When this occurs, the figure rotates 80. So, when k > 0, a dilation with a scale factor of k is the same as the composition of a dilation with a scale factor of k followed b a rotation of 80 about the center of dilation. Using the coordinate rules for a dilation and a rotation of 80, ou can think of the notation as (, ) (k, k) ( k, k). Using a Negative Scale Factor scale factor k Graph FGH with vertices F(, ), G(, ), and H(, ) and its image after a dilation with a scale factor of. Use the coordinate rule for a dilation with k = to find the coordinates of the vertices of the image. Then graph FGH and its image. (, ) (, ) F(, ) F (, ) G(, ) G (, ) H(, ) H (, ) G H F F H G 0 hapter Transformations Monitoring rogress Help in English and Spanish at BigIdeasMath.com. Graph Q with vertices (, ), Q(, ), and (, ) and its image after a dilation with a scale factor of. 5. Suppose a figure containing the origin is dilated. Eplain wh the corresponding point in the image of the figure is also the origin.

5 EDING Scale factors are written so that the units in the numerator and denominator divide out. Solving eal-life roblems Finding a Scale Factor You are making our own photo stickers. Your photo is inches b inches. The image on the stickers is. inches b. inches. What is the scale factor of this dilation? The scale factor is the ratio of a side length of the sticker image to a side. in. length of the original photo, or in.. So, in simplest form, the scale factor is 0. Finding the Length of an Image You are using a magnifing glass that shows the image of an object that is si times the object s actual size. Determine the length of the image of the spider seen through the magnifing glass. image length actual length = k.5 = 6 = 9. in. in..5 cm So, the image length through the magnifing glass is 9 centimeters..6 cm Monitoring rogress Help in English and Spanish at BigIdeasMath.com 6. n optometrist dilates the pupils of a patient s ees to get a better look at the back of the ees. pupil dilates from.5 millimeters to 8 millimeters. What is the scale factor of this dilation? 7. The image of a spider seen through the magnifing glass in Eample 6 is shown at the left. Find the actual length of the spider. When a transformation, such as a dilation, changes the shape or size of a figure, the transformation is nonrigid. In addition to dilations, there are man possible nonrigid transformations. Two eamples are shown below. It is important to pa close attention to whether a nonrigid transformation preserves lengths and angle measures. Horizontal Stretch Vertical Stretch B B B Section.5 Dilations

6 .5 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and ore oncept heck. OMLETE THE SENTENE If (, ) is the preimage of a point, then its image after a dilation centered at the origin (0, 0) with scale factor k is the point.. WHIH ONE DOESN T BELONG? Which scale factor does not belong with the other three? Eplain our reasoning. 5 60% 5% Monitoring rogress and Modeling with Mathematics In Eercises 6, find the scale factor of the dilation. Then tell whether the dilation is a reduction or an enlargement. (See Eample.) ONSTUTION In Eercises, cop the diagram. Then use a compass and straightedge to construct a dilation of quadrilateral STU with the given center and scale factor k. S U. enter, k =. enter, k =. enter, k = 0.5. enter, k = 75% T ONSTUTION In Eercises 7 0, cop the diagram. Then use a compass and straightedge to construct a dilation of LMN with the given center and scale factor k. L In Eercises 5 8, graph the polgon and its image after a dilation with scale factor k. (See Eamples and.) 5. X(6, ), Y(, ), Z(, ); k = 6. (0, 5), B( 0, 5), (5, 5); k = 0% 7. T(9, ), U(6, 0), V(, 9), W(0, 0); k = M 8. J(, 0), K( 8, ), L(0, ), M(, 8); k = enter, k = 8. enter, k = 9. enter M, k = 0. enter, k = 5% hapter Transformations N In Eercises 9, graph the polgon and its image after a dilation with scale factor k. (See Eample.) 9. B( 5, 0), ( 0, 5), D(0, 5); k = 5 0. L(0, 0), M(, ), N(, 6); k =. ( 7, ), S(, 5), T(, ), U(, ); k =. W(8, ), X(6, 0), Y( 6, ), Z(, ); k = 0.5

7 EO NLYSIS In Eercises and, describe and correct the error in finding the scale factor of the dilation.. k = = In Eercises, ou are using a magnifing glass. Use the length of the insect and the magnification level to determine the length of the image seen through the magnifing glass. (See Eample 6.). emperor moth. ladbug Magnification: 5 Magnification: 0. k = = In Eercises 5 8, the red figure is the image of the blue figure after a dilation with center. Find the scale factor of the dilation. Then find the value of the variable (, ) (, ) n 8 60 mm.5 mm. dragonfl. carpenter ant Magnification: 0 Magnification: 5 7 mm mm 5. NLYZING ELTIONSHIS Use the given actual and magnified lengths to determine which of the following insects were looked at using the same magnifing glass. Eplain our reasoning. grasshopper black beetle ctual: in. ctual: 0.6 in. Magnified: 5 in. Magnified:. in m 7 8 honebee monarch butterfl ctual: 5 in. 8 ctual:.9 in. Magnified: 75 in. 6 Magnified: 9.5 in. 9. FINDING SLE FTO You receive wallet-sized photos of our school picture. The photo is.5 inches b.5 inches. You decide to dilate the photo to 5 inches b 7 inches at the store. What is the scale factor of this dilation? (See Eample 5.) 0. FINDING SLE FTO Your visuall impaired friend asked ou to enlarge our notes from class so he can stud. You took notes on 8.5-inch b -inch paper. The enlarged cop has a smaller side with a length of 0 inches. What is the scale factor of this dilation? (See Eample 5.) 6. THOUGHT OVOKING Draw B and B so that B is a dilation of B. Find the center of dilation and eplain how ou found it. 7. ESONING Your friend prints a -inch b 6-inch photo for ou from the school dance. ll ou have is an 8-inch b 0-inch frame. an ou dilate the photo to fit the frame? Eplain our reasoning. Section.5 Dilations

8 8. HOW DO YOU SEE IT? oint is the center of dilation of the images. The scale factor is. Which figure is the original figure? Which figure is the dilated figure? Eplain our reasoning. 9. MTHEMTIL ONNETIONS The larger triangle is a dilation of the smaller triangle. Find the values of and. ( ) 5. NLYZING ELTIONSHIS Dilate the line through O(0, 0) and (, ) using a scale factor of. a. What do ou notice about the lengths of O and O? b. What do ou notice about O and O? 6. NLYZING ELTIONSHIS Dilate the line through (0, ) and B(, ) using a scale factor of. a. What do ou notice about the lengths of B and B? b. What do ou notice about B and B? 7. TTENDING TO EISION You are making a blueprint of our house. You measure the lengths of the walls of our room to be feet b feet. When ou draw our room on the blueprint, the lengths of the walls are 8.5 inches b 9 inches. What scale factor dilates our room to the blueprint? ( + 6) MKING N GUMENT Your friend claims that dilating a figure b is the same as dilating a figure b because the original figure will not be enlarged or reduced. Is our friend correct? Eplain our reasoning WITING Eplain wh a scale factor of is the same as 00%. In Eercises, determine whether the dilated figure or the original figure is closer to the center of dilation. Use the given location of the center of dilation and scale factor k.. enter of dilation: inside the figure; k =. enter of dilation: inside the figure; k = 9. USING STUTUE ectangle WXYZ has vertices W(, ), X(, ), Y(5, ), and Z(5, ). a. Find the perimeter and area of the rectangle. b. Dilate the rectangle using a scale factor of. Find the perimeter and area of the dilated rectangle. ompare with the original rectangle. What do ou notice? c. epeat part (b) using a scale factor of. d. Make a conjecture for how the perimeter and area change when a figure is dilated. 50. ESONING You put a reduction of a page on the original page. Eplain wh there is a point that is in the same place on both pages.. enter of dilation: outside the figure; k = 0%. enter of dilation: outside the figure; k = 0. Maintaining Mathematical roficienc 5. ESONING B has vertices (, ), B(, 6), and (7, ). Find the coordinates of the vertices of the image after a dilation with center (, 0) and a scale factor of. eviewing what ou learned in previous grades and lessons The vertices of B are (, ), B(0, ), and (, 5). Find the coordinates of the vertices of the image after the translation. (Section.) 5. (, ) (, ) 5. (, ) (, + ) 5. (, ) ( +, ) 55. (, ) (, ) 56. (, ) ( +, ) 57. (, ) (, + ) hapter Transformations