Investigating Properties of Dilations

Transcription

1 Name lass Date 1.1 Dilations Essential Question: How does a dilation transform a figure? Eplore 1 Investigating Properties of Dilations dilation is a transformation that can change the size of a polgon but leaves the shape unchanged. dilation has a center of dilation and a scale factor which together determine the position and size of the image of a figure after the dilation. Resource Locker Use and its image ''' after a dilation to answer the following questions. ' ' ' Use a ruler to measure the following lengths. Measure to the nearest tenth of a centimeter. Use a protractor to measure the corresponding angles. Houghton Mifflin Harcourt Publishing ompan = cm '' = cm = cm '' = cm = cm '' = cm omplete the following ratios m = m ' = m = m ' = m = m ' = _ '' = _ = _ '' = _ = _ '' = _ = Reflect 1. What do ou notice about the corresponding sides of the figures? What do ou notice about the corresponding angles?. Discussion What similarities are there between reflections, translations, rotations, and dilations? What is the difference? Module 1 87 Lesson 1

2 Eplore Dilating a Line Segment The dilation of a line segment (the pre-image) is a line segment whose length is the product of the scale factor and the length of the pre-image. Use the following steps to appl a dilation b a factor of 3, with center at the point, to. D To locate the point ', draw a ra from through. Place ' on this ra so that the distance from to ' is three times the distance from to. To locate point, draw a ra from through. Place on this ra so that the distance from to is three times the distance from to. To locate point, draw a ra from through. Place on this ra so that the distance from to is three times the distance from to. Draw a line through ', ', and '. E Measure _, _, and _. Measure _ '', _ '', and _ ''. Make a conjecture about the lengths of segments that have been dilated. Reflect 3. Make a conjecture about the length of the image of a cm segment after a dilation with scale factor k. an the image ever be shorter than the preimage?. What can ou sa about the image of a segment under a dilation? Does our answer depend upon the location of the segment? Eplain Houghton Mifflin Harcourt Publishing ompan Module 1 88 Lesson 1

3 Eplain 1 ppling Properties of Dilations The center of dilation is the fied point about which all other points are transformed b a dilation. The ratio of the lengths of corresponding sides in the image and the preimage is called the scale factor. Properties of Dilations Dilations preserve angle measure. Dilations preserve betweenness. Dilations preserve collinearit. Dilations preserve orientation. Dilations map a line segment (the pre-image) to another line segment whose length is the product of the scale factor and the length of the pre-image. Dilations map a line not passing through the center of dilation to a parallel line and leave a line passing through the center unchanged. Eample 1 Determine if the transformation on the coordinate plane is a dilation. If it is, give the scale factor. Preserves angle measure: es Preserves betweenness: es Preserves collinearit: es D' ' D Preserves orientation: no Ratio of corresponding sides: 1 : 1 Is this transformation a dilation? No, it does not preserve orientation. ' ' Preserves angle measure (Y/N) ' Houghton Mifflin Harcourt Publishing ompan Preserves betweenness (Y/N) Preserves collinearit (Y/N) Preserves orientation (Y/N) Scale Factor Is this transformation a dilation? - ' ' Module 1 89 Lesson 1

4 Your Turn Determine if the transformations are dilations. 5. ' ' - - ' D' D - - E E'. -8 ' - - ' - - ' - Eplain Determining the enter and Scale of a Dilation When ou have a figure and its image after dilation, ou can find the center of dilation b drawing lines that connect corresponding vertices. These lines will intersect at the center of dilation. Eample Determine the center of dilation and the scale factor of the dilation of the triangles. Draw ', ', and '. The point where the lines cross is the center of dilation. Label the intersection. Measure to find the scale factor. = 5 mm = 13 mm = 19 mm = 5 mm = mm = 38 mm The scale factor is to 1. ' ' ' Houghton Mifflin Harcourt Publishing ompan Module 1 83 Lesson 1

5 Draw ', ', and '. Measure from each point to the intersection to the nearest millimeter. = = = ' = ' = = The scale factor is. ' Reflect 7. For the dilation in Your Turn 5, what is the center of dilation? Eplain how ou can tell without drawing lines. Your Turn 8. Determine the center of dilation and the scale factor of the dilation. Houghton Mifflin Harcourt Publishing ompan ' = cm, = The scale factor of the dilation is. Elaborate 9. How is the length of the image of a line segment under a dilation related to the length of its preimage? 1. Discussion What is the result of dilating a figure using a scale factor of 1? For this dilation, does the center of dilation affect the position of the image relative to the preimage? Eplain. ' ' ' Module Lesson 1

6 11. Essential Question heck-in In general how does a dilation transform a figure? Evaluate: Homework and Practice 1. onsider the definition of a dilation. dilation is a transformation that can change the size of a polgon but leaves the shape unchanged. In a dilation, how are the ratios of the measures of the corresponding sides related? nline Homework Hints and Help Etra Practice Tell whether one figure appears to be a dilation of the other figure Eplain Is the scale factor of the dilation of equal to 1 _? Eplain. ' ' 5. Square is a dilation of square. What is the scale factor? a. 1_ 7 b. _ 5 c. 5_ d. 7 e. _ ' 8 8 Houghton Mifflin Harcourt Publishing ompan Module 1 83 Lesson 1

7 . ppl a dilation to _ with a scale factor of and center at the point. 7. ppl a dilation to _ with a scale factor of 1 _ 3 and center at the point. 8. What happens when a triangle is dilated using one of the vertices as the center of dilation? 9. Draw an image of WXYZ. The center of the dilation is, and the scale factor is. X Y W Z 1. Draw an image of. The center of dilation is, and the scale factor is ompare dilations to rigid motions. How are the the same? How are the different? Houghton Mifflin Harcourt Publishing ompan Determine if the transformation of figure to figure on the coordinate plane is a dilation. Verif ratios of corresponding side lengths for a dilation Module Lesson 1

8 Determine the center of dilation and the scale factor of the dilation ' E ' ' D E' F D' F' The scale factor is. The scale factor is. 1. You work at a photograph store. customer has a picture that is.5 inches tall. The customer wants a reduced cop of the picture to fit a space of 1.8 inches tall on a postcard. What scale factor should ou use to reduce the picture to the correct size? 17. omputer Graphics n artist uses a computer program to enlarge a design, as shown. What is the scale factor of the dilation? (, ) '(, 1) '(15, 1) '(, ) (5, ) D'(15, ) D(5, ) (, ) Houghton Mifflin Harcourt Publishing ompan Image redits: Digital Vision/Gett Images Module 1 83 Lesson 1

9 18. Eplain the Error What mistakes did the student make when tring to determine the center of dilation? Determine the center of dilation. P P' R Q R' Q' H..T. Focus on Higher rder Thinking Draw DEF with vertices D (3, 1) E (3, 5) F (, 5). 1 a. Determine the perimeter and the area of DEF. 1 b. Draw an image of DEF after a dilation having a scale factor of 3, with the center of dilation at the origin (, ). Determine the perimeter and area of the image. 8 perimeter D'E'F' c. How is the scale factor related to the ratios perimeter DEF area D'E'F' and area DEF? 8 Houghton Mifflin Harcourt Publishing ompan. Draw WXY with vertices (, ), (, 8), and (-, 8). a. Dilate WXY using a factor of 1_ and the origin as the center. Then dilate its image using a scale factor of and the origin as the center. Draw the final image. b. Use the scale factors given in part (a) to determine the scale factor ou could use to dilate WXY with the origin as the center to the final image in one step. c. Do ou get the same final image if ou switch the order of the dilations in part (a)? Eplain our reasoning. - 8 Module Lesson 1

10 Lesson Performance Task You ve hung a sheet on a wall and lit a candle. Now ou move our hands into position between the candle and the sheet and, to the great amusement of our audience, create an image of an animal on the sheet. ompare and contrast what ou re doing with what happens when ou draw a dilation of a triangle on a coordinate plane. Point out was that dilations and hand puppets are alike and was the are different. Discuss measures that are preserved in hand-puppet projections and those that are not. Some terms ou might like to discuss: pre-image image center of dilation scale factor transformation input output Houghton Mifflin Harcourt Publishing ompan Image redits: Digital Vision/Gett Images Module 1 83 Lesson 1