Representations of Dilations 8.3.C

Transcription

1 ? LESSN 1. lgebraic Representations of Dilations ESSENTIL QUESTIN Proportionalit.. 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..,.10.d How can ou describe the effect of a dilation on coordinates using an algebraic representation? EXPLRE TIVITY 1.. 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 (, ) 7 (, ) (, ) Houghton Mifflin Harcourt Publishing ompan D 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 1. 9

2 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.. 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_ (, ) 5 What is the scale factor for the dilation? - D 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. How does the dilation affect the length of line segments? -5 Houghton Mifflin Harcourt Publishing ompan. How would a dilation with scale factor 1 affect the preimage? 70 Unit 5

3 enter of Dilation 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 1.. Graph the image of after a dilation with the origin as its center and a scale factor of. What are the vertices of the image? STEP 1 Multipl each coordinate of the vertices of b to find the vertices of the dilated image. (, ) (, ) (1, 1) (1, 1 ) (, ) (, 1) (, 1 ) (9, ) (1, ) (1, ) (, 9) The vertices of the dilated image are (, ), (9, ), and (, 9). STEP Graph the dilated image. ' Math Talk Mathematical Processes Houghton Mifflin Harcourt Publishing ompan YUR TURN ' 5. Graph the image of XYZ after a dilation with a scale factor of 1_ and the origin as its center. Then write an algebraic rule to describe the dilation. ' Describe how ou can check graphicall that ou have drawn the image triangle correctl. Z X Y Personal Math Trainer nline ssessment and Intervention m.hrw.com Lesson 1. 71

4 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 (, ) ( _, _ ) 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) (, 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 1.5. scale factor of 1_ I H F G? ESSENTIL QUESTIN HEK-IN. 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 7 Unit 5

5 LESSN 1. Dilations and Measurement Two-dimensional shapes.10.d Model the effect on linear and area measurements of dilated two-dimensional shapes. lso..,.10.,.10.? ESSENTIL QUESTIN How do ou describe the effects of dilation on linear and area measurements? EXPLRE TIVITY.10.D Eploring Dilations and Measurement The blue rectangle is a dilation (enlargement) of the green rectangle. Using a centimeter ruler, measure and record the length of each side of both rectangles. Then calculate the ratios of all pairs of corresponding sides. ' D ' = = D = D = '' = '' = 'D' = D'' = D' ' '' = '' = 'D' D = What is true about the ratios that ou calculated? D'' D = Houghton Mifflin Harcourt Publishing ompan What scale factor was used to dilate the green rectangle to the blue rectangle? How are the side lengths of the blue rectangle related to the side lengths of the green rectangle? What is the perimeter of the green rectangle? What is the perimeter of the blue rectangle? How is the perimeter of the blue rectangle related to the perimeter of the green rectangle? Lesson 1. 75

6 EXPLRE TIVITY (cont d) What is the area of the green rectangle? What is the area of the blue rectangle? How is the area of the blue rectangle related to the area of the green rectangle? Reflect 1. Make a onjecture The perimeter and area of two shapes before and after dilation are given. How are the perimeter and area of a dilated figure related to the perimeter and area of the original figure? Perimeter rea riginal Dilation 1 1 Perimeter rea riginal 0 5 Dilation Scale factor = Scale factor = 1_ Math n the Spot m.hrw.com nimated Math m.hrw.com Problem-Solving pplication Understanding how dilations affect the linear and area measurements of shapes will enable ou to solve man real-world problems. EXMPLE 1 Problem Solving souvenir shop sells standard-sized decks of cards and mini-decks of cards. card in the standard deck is a rectangle that has a length of.5 inches and a width of.5 inches. The perimeter of a card in the mini-deck is inches. What is the area of a card in the mini-deck? nalze Information I need to find the area of a mini-card. I know the length and width of a standard card and the perimeter of a mini-card... Houghton Mifflin Harcourt Publishing ompan Image redits: Stockbte/ Gett Images 7 Unit 5

7 Formulate a Plan M Notes Since the mini-card is a dilation of the standard card, the figures are similar. Find the perimeter of the standard card, and use that to find the scale factor. Then use the scale factor to find the area of the mini-card. Justif Solve and Evaluate STEP 1 Find the perimeter of the standard card. P s = l + w P s = (.5) + (.5) P s = 1 in. STEP Find the scale factor. P m = P s k = 1 k 1_ = k Multipl the perimeter of the standard card b the scale factor to get the perimeter of the mini-card. STEP Find the area of the standard card. s = l s w s s =.5.5 Use the formula for the area of a rectangle. s =.75 in STEP Find the area of the mini-card. Houghton Mifflin Harcourt Publishing ompan m = s k m =.75 ( 1_ ) m =.75 1_ m =.175 in Multipl the area of the standard card b the scale factor squared to get the area of the mini-card. The area of the mini card is about. square inches. Justif and Evaluate To find the area of the mini-card, find its length and width b multipling the dimensions of the standard card b the scale factor. The length of the mini-card is l s 1_ =.5 1_ = 1.75 in., and the width is w s 1_ =.5 1_ = 1.5 in. So, m = l m w m = =.175 in. The answer is correct. Lesson 1. 77

8 YUR TURN Personal Math Trainer nline ssessment and Intervention m.hrw.com. Johnson Middle School is selling mouse pads that are replicas of a student s award-winning artwork. The rectangular mouse pads are dilated from the original artwork and have a length of 9 inches and a width of inches. The perimeter of the original artwork is 1 inches. What is the area of the original artwork? Guided Practice Find the perimeter and area of the image after dilating the figures shown with the given scale factor. (Eplore ctivit and Eample 1) 1. Scale factor = 5. Scale factor = _ 1 P = = 1 P = 1 = 9 P' = ' = P' = ' = group of friends is roping off a soccer field in a back ard. full-size soccer field is a rectangle with a length of 100 ards and a width of 0 ards. To fit the field in the back ard, the group needs to reduce the size of the field so its perimeter is 1 ards. (Eample 1)?. What is the perimeter of the full-size soccer field?. What is the scale factor of the dilation? 5. What is the area of the soccer field in the back ard? ESSENTIL QUESTIN HEK-IN. When a rectangle is dilated, how do the perimeter and area of the rectangle change? Houghton Mifflin Harcourt Publishing ompan 7 Unit 5