Dilations and Measurement 8.10.D. Exploring Dilations and Measurement The blue rectangle is a dilation (enlargement) of the green rectangle.
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1 LESSON 13.3 Dilations and Measurement Two-dimensional shapes 8.10.D Model the effect on linear and area measurements of dilated two-dimensional shapes. Also 8.3.B, 8.10.A, 8.10.B? ESSENTIAL QUESTION How do you describe the effects of dilation on linear and area measurements? EXPLORE ACTIVITY 8.10.D Exploring Dilations and Measurement The blue rectangle is a dilation (enlargement) of the green rectangle. A 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. A' A D B C B' AB = BC = CD = DA = A'B' = B'C' = C'D' = D'A' = D' C' A'B' AB = B'C' BC = C'D' CD = What is true about the ratios that you calculated? D'A' DA = B 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
2 EXPLORE ACTIVITY (cont d) C 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 Conjecture 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 Area Original 8 4 Dilation Perimeter Area Original Dilation Scale factor = 2 Scale factor = 1_ 6 Math On the Spot Animated Math Problem-Solving Application Understanding how dilations affect the linear and area measurements of shapes will enable you to solve many real-world problems. EXAMPLE 1 Problem Solving A souvenir shop sells standard-sized decks of cards and mini-decks of cards. A card in the standard deck is a rectangle that has a length of 3.5 inches and a width of 2.5 inches. The perimeter of a card in the mini-deck is 6 inches. What is the area of a card in the mini-deck? Analyze 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. 8.3.B Image Credits: Stockbyte/ Getty Images 376 Unit 5
3 Formulate a Plan My 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. Justify Solve and Evaluate STEP 1 Find the perimeter of the standard card. P s = 2l + 2w P s = 2(3.5) + 2(2.5) P s = 12 in. STEP 2 Find the scale factor. P m = P s k 6 = 12 k 1_ 2 = k Multiply the perimeter of the standard card by the scale factor to get the perimeter of the mini-card. STEP 3 Find the area of the standard card. A s = l s w s A s = Use the formula for the area of a rectangle. A s = 8.75 in 2 STEP 4 Find the area of the mini-card. A m = A s k 2 A m = 8.75 ( 1_ 2) 2 A m = _ 4 A m = in 2 Multiply the area of the standard card by the scale factor squared to get the area of the mini-card. The area of the mini card is about 2.2 square inches. Justify and Evaluate To find the area of the mini-card, find its length and width by multiplying the dimensions of the standard card by the scale factor. The length of the mini-card is l s 1_ 2 = 3.5 1_ 2 = 1.75 in., and the width is w s 1_ 2 = 2.5 1_ 2 = 1.25 in. So, A m = l m w m = = in 2. The answer is correct. Lesson
4 YOUR TURN Personal Math Trainer Online Assessment and Intervention 2. 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 8 inches. The perimeter of the original artwork is 136 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. (Explore Activity and Example 1) 1. Scale factor = Scale factor = 3 _ P = 48 A = 128 P = 12 A = 9 P' = A' = P' = A' = A group of friends is roping off a soccer field in a back yard. A full-size soccer field is a rectangle with a length of 100 yards and a width of 60 yards. To fit the field in the back yard, the group needs to reduce the size of the field so its perimeter is 128 yards. (Example 1)? 3. What is the perimeter of the full-size soccer field? 4. What is the scale factor of the dilation? 5. What is the area of the soccer field in the back yard? ESSENTIAL QUESTION CHECK-IN 6. When a rectangle is dilated, how do the perimeter and area of the rectangle change? 378 Unit 5
5 Name Class Date 13.3 Independent Practice 8.3.B, 8.10.A, 8.10.B, 8.10.D Personal Math Trainer Online Assessment and Intervention 7. When you make a photocopy of an image, is the photocopy a dilation? What is the scale factor? How do the perimeter and area change? 8. Problem Solving The universally accepted film size for movies has a width of 35 millimeters. If you want to project a movie onto a square sheet that has an area of 100 square meters, what is the scale factor that is needed for the projection of the movie? Explain. 9. The perimeter of a square is 48 centimeters. If the square is dilated by a scale factor of 0.75, what is the length of each side of the new square? 10. The screen of an ereader has a length of 8 inches and a width of 6 inches. Can the page content from an atlas that measures 19 inches by 12 inches be replicated in the ereader? If not, propose a solution to move the atlas content into the ereader format. 11. Represent Real-World Problems There are 64 squares on a chessboard. Each square on a tournament chessboard measures inches. A travel chessboard is a dilated replica of the tournament chessboard using a scale factor of 1_ 3. a. What is the size of each square on the travel chessboard? b. How long is each side of the travel board? c. How much table space do you need to play on the travel chessboard? Lesson
6 12. Draw Conclusions The legs of a right triangle are 3 units and 4 units long. Another right triangle is dilated from this triangle using a scale factor of 3. What are the side lengths and the perimeter of the dilated triangle? FOCUS ON HIGHER ORDER THINKING Work Area 13. Critique Reasoning Rectangle W X Y Z below is a dilation of rectangle WXYZ. A student calculated the area of rectangle W'X'Y'Z' to be 36 square units. Do you agree with this student's calculation? If not, explain and correct the mistake. W 12 X W' 4 X' 9 Z' Y' Z Y 14. Multistep Rectangle A'B'C'D' is a dilation of rectangle ABCD, and the scale factor is 2. The perimeter of ABCD is 18 mm. The area of ABCD is 20 mm 2. a. Write an equation for, and calculate, the perimeter of A'B'C'D'. b. Write an equation for, and calculate, the area of A'B'C'D'. c. The side lengths of both rectangles are whole numbers of millimeters. What are the side lengths of ABCD and A'B'C'D'? 380 Unit 5
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