BLoCK 2 ~ ProPortIonALItY

Transcription

1 BLoCK 2 ~ ProPortIonALItY PrOPOrtiOns and similarity Lesson 6 write and solve ProPorTions Explore! Macaroni and Cheese Lesson 7 ProBLeM solving with ProPorTions Explore! Cookies Lesson 8 similar and congruent FigUres Explore! Similar Triangles Lesson 9 ProPorTions and similar FigUres Lesson 10 special ratios For similar FigUres Explore! Perimeter and Area Lesson 11 scale drawings review BLock 2 ~ ProPorTions and similarity word wall scale FActor ProPortion similar Figures congruent Figures congruent corresponding PArts 32 Block 2 ~ Proportionality ~ Proportions And Similarity

2 BLoCK 2 ~ ProPortIons And similarity tic - tac - toe money conversions gifts how tall? Convert currencies in other countries to dollars. Two similar presents need ribbon and wrapping paper. Find how much. Use proportions to find the height of two objects. See page 42 for details. See page 59 for details. See page 53 for details. contractor home Floor PlAn children s story Explain the mistake a contractor made when buying paint at the store. Draw a floor plan for a single story home. Write a children s story involving proportions. See page 69 for details. See page 64 for details. See page 59 for details. map cookies Picture sizes Use a local map to find actual distances. Use a recipe for chocolate chip cookies to increase and decrease the amount of cookies it makes. Are picture sizes all similar? Look at two pictures of yourself to find out. See page 38 for details. See page 49 for details. See page 49 for details. Block 2 ~ Proportions And Similarity ~ Tic - Tac - Toe 33

3 write and solve PrOPOrtiOns Lesson 6 explore! macaroni nd aheese c Ahn and four friends decided to make macaroni & cheese. They were very hungry so Ahn decided to make two boxes instead of one. The directions for one box are step 1: Find the amount of milk, butter and cheese packets Ahn will need to make both boxes of macaroni and cheese. step 2: Write the simplified ratio of the amount of milk needed for one box to the amount of milk needed for two boxes. step 3: Write the simplified ratio of the amount of butter needed for one box to the amount of butter needed for two boxes. step 4: Write the simplified ratio of the number of cheese packets needed for one box to the number of cheese packets needed for two boxes. step 5: What ratio did you write for steps #2-4? Explain the meaning of the ratio. step 6: Explain the meaning of a ratio of 1 : 3 in a similar situation. step 7: The next day Ahn used 8 tablespoons of butter to make macaroni and cheese. a. How many boxes of macaroni and cheese did he make? b. How much milk did he use? In the Explore! Ahn first made two boxes of macaroni and cheese. The ratio of the amount of milk used in one box compared to two boxes was equal to the ratio of butter used in one box compared to two boxes. Since the ratios are equal, they can be written with an equals sign. 34 Lesson 6 ~ Write And Solve Proportions 4_ 8 = _ 2 4 This is a proportion. A proportion is an equation stating that two ratios are equal. Proportions have equal cross products. Cross products are calculated by multiplying the numerator of one ratio by the denominator of the other ratio. Each cross product is placed on one side of an equals sign. For example: 4_ 8 = _ = = 16

4 When one number in a proportion is missing, you can find it using equivalent fractions or cross products. Suppose Ahn tripled the amount of macaroni and cheese. Each of his ingredients from one box to three boxes will be in a ratio of 1 : 3. Use a proportion to solve for the amount of milk Ahn will need. solving Proportions Method 1 ~ equivalent Fractions Write a proportion. 1_ 3 = _ 4 x Figure out what to multiply 1 by to get 4. Multiply the denominator by this number. 1_ Multiply to get the equivalent fraction. x = 12 = 4_ x Method 2 ~ Cross Products Write a proportion. 1_ 3 = _ 4 x Write the cross products. 1 x = 4 3 Simplify. 1x = 12 Solve for x by dividing both sides of the equation by 1. x = 12 example 1 solve each proportion using cross products. a. 2 3 = x b a = solutions a. Write the cross products. 3 x = 2 15 Simplify the equation. 3x = 30 Divide both sides by 3. x = 10 Check the answer. Substitute 10 for x. 2_ 3 = simplifies to 2_ 3, so the solution is correct. b. Write the cross products. 1.5 a = 6 7 Simplify. 1.5a = 42 Divide both sides by 1.5. a = 28? Check the answer =? (28) =? = 42 Lesson 6 ~ Write And Solve Proportions 35

5 example 2 determine whether each pair of ratios forms a proportion. a. 4 9 and 2 6 b and solutions a. Write the proportion using? =. 4_ 9 =? _ 2 6? Check using cross products. 4 6 = No, 4_ 9 and 2_ 6 do not form a proportion. The cross products are not equal, so the ratios are not equivalent. b. Write the proportion using? = =? 50 30? Find the cross products = = 450 Yes, and 30 do form a proportion. Their cross products are equal. They are equivalent ratios. example 3 Write a proportion for the following phrase and solve it. $4 in 30 minutes; $b in 45 minutes solution If the first ratio is written as dollars in the numerator and minutes in the denominator, the second must be written the same way. Write the cross products. Simplify. Divide both sides of the equation by 30. $6 in 45 minutes. $4 30 minutes = $b 45 minutes 4 45 = 30 b 180 = 30b = 30b 30 6 = b 36 Lesson 6 ~ Write And Solve Proportions

6 exercises solve each proportion. 1. 2_ 3 = x _ b = _ 7. 3 = 21 x 2. 5_ 3 = 45 x 5. 5_ 8 = x a 17 = a 12 = 2 24 y 10 = = _ x 6 determine whether each pair of ratios forms a proportion _ 3 and _ 8 and _ 4 and and and and use cross products to check the solution given for each proportion. Write yes or no for whether or not the solution works. explain by showing your work _ 4 = 15 x ; x = _ 9 = 24 x ; x = _ 5 = a 12 ; a = 4.8 Write a proportion for each phrase and solve it miles in 2 hours; x miles in 6 hours minutes for 3 typed pages; 60 minutes for a typed pages pounds for $10.95; y pounds for $ feet in 4 seconds; 80 feet in x seconds 23. Larry paid $15 for 6 boxes of crackers. Nancy bought 4 boxes of the same crackers at another store. She paid $11. a. Write Larry s ratio of the cost to the number of boxes of crackers. b. Write Nancy s ratio of the cost to the number of boxes of crackers. c. Do the ratios form a proportion? Why or why not? Lesson 6 ~ Write And Solve Proportions 37

7 24. Paulo and Sami solved the proportion = x 45. Their work is shown at right. a. Determine whether or not they answered the question correctly. b. Why does Paulo multiply by 3 in the numerator and denominator in step 1? c. Sami wanted to use smaller numbers in her proportion. What did she do for step 1 to create smaller numbers? d. Which method do you like best? Why? review 1) Paulo s Work = x = x 45 2) x = 36 1) sami s Work = x 45 4_ 5 = x 45 2) 5x = 180 3) x = 36 Find each unit rate balloons $ miles 1.5 hours miles 5 gallons 20 ft 12 seconds 29. A store sells 2 pair of shoes for $ Find the price for each pair of shoes. 30. A car travels 204 miles using 12 gallons of gas. Find the miles per gallon the car used. tic-tac-toe ~ map 1. Print or copy a local map that includes your home and school. Include the map with your work. 2. Record the scale of the map (map length : actual length). 3. Use a red pen to trace along the streets from your home to your school. Use string along the path in red to find the distance on the map from your home to your school. Record the length of the string. 4. Use proportions to find the actual distance from your home to your school. 5. Trace a path in blue between two places on your map and a path in green between two places on your map. (You can use any color, just different from the ones already used.) 6. Use string along each path in Part 5 to find the map distance. Record the map distance. 7. Use proportions to find the actual distance for each path. 8. Research how far it is from your home to the Portland Zoo. How many inches would represent that distance on your map? 38 Lesson 6 ~ Write And Solve Proportions

8 PrOBlem solving with PrOPOrtiOns Lesson 7 Many real-world situations are solved using proportions. When setting up proportions, it is very important to place values correctly in the proportion. explore! cookies Ethan and Priscilla bought cookies at the store. Ethan bought two dozen cookies for $4.80. Priscilla bought 5 dozen of the same cookies. step 1: Find the price of the cookies per dozen (the unit rate). step 2: Use the unit rate to determine the amount Priscilla paid for five dozen cookies. step 3: Write a ratio comparing the total cost of the cookies to the number of dozens purchased for Ethan. Write a similar ratio for Priscilla s purchase. step 4: Set the ratios equal. Are the cross products equal? step 5: One week later, Ethan went back to the store to buy eight dozen of the same cookies. The cookies were the same price per dozen. Ethan used a proportion to determine the cost for eight dozen cookies. Look at the proportions below. a. Which one could be used to find the cost of the eight dozen cookies? b. Why does the other one not work? $ dozen = $x 8 dozen $ dozen = 8 dozen $x step 6: Solve the proportion chosen in step 5a to find the cost of 8 dozen cookies. step 7: Use the unit rate for the cookies found in step 1. Multiply the unit rate by 8. Does this match the answer to step 6? step 8: Ethan also bought 3 pounds of apples for $3.90. Priscilla bought 5 pounds of the same type of apples. Set up a proportion to find the price Priscilla paid for her apples. Verify the answer using unit rates. Lesson 7 ~ Problem Solving With Proportions 39

9 example 1 solution students help run a summer soccer camp. two students are needed for every 15 elementary school soccer players. There are 195 soccer players signed up. how many students are needed? Let x = the number of students needed. Write a proportion. 2 students 15 players = x students 195 players 2 15 = x 195 Write the cross products. 15x = 390 Divide by 15. x = 26 Twenty-six students are needed for soccer camp. example 2 solution Mandy read 180 pages in 4 hours. how long will it take her to read a 225 page book at this rate? Let x = the number of hours it takes Mandy to read a 225 page book. Write a proportion. 180 pages 4 hours Write the cross products. 180x = 900 Divide by 180. x = 5 It will take Mandy 5 hours to read a 225 page book. = 225 pages x hours 40 Lesson 7 ~ Problem Solving With Proportions

10 exercises solve each proportion = 4 _ a solve each problem using a proportion = _ b = _ x 9 4. Peggy rode her bike 25 miles in 2 hours. At this rate, how far will she ride in 3 hours? 5. Camden went bowling with some friends. It cost $12.00 to rent 10 pairs of shoes. Fifteen people were bowling. How much would it cost him to pay for 15 pairs of shoes? 6. Sergio ate two cookies which contained a total of 14 chocolate chips. His friend, Evan, exclaimed, Wow. I ate 49 chocolate chips! Every cookie has the same number of chocolate chips. How many cookies did Evan eat? 7. Kendrey ran 9 miles in 1 hour and 30 minutes. How long will it take her to run 12 miles at this pace? 8. Willen decided to shave his head with the rest of his baseball team once they made the playoffs. Hair grows at a rate of 0.03 inches per day. How long will it take his hair to grow 3 inches? 9. Carl finished 2 holes in 25 minutes when playing golf. At this rate, how long will it take Carl to play all 18 holes? 10. Dialo walked 42,240 feet in two hours. How many miles will he walk in three hours at this speed? 11. It takes 5 bags of potting soil to cover Margaret s garden. Next year she plans to increase the size of her garden from 64 square feet to 96 square feet. How many bags of potting soil will she need to cover her new garden? 12. It costs $1.60 for a 32 ounce soda at a local market. It costs $1.92 for a 48 ounce soda. a. Find the cost per ounce for the 32 ounce soda. b. Find the cost per ounce for the 48 ounce soda. c. Which soda is the better deal? (This means cheapest per ounce of soda.) d. What would be the cost of the 48 ounce soda if it is sold at the same rate as the 32 ounce soda? 13. At Movies-4-Us it costs $20 to buy 3 movies or $25 to buy 4 movies. a. Find the cost per movie for both options. b. Which option is the better deal? (This means cheapest price per movie.) c. How much would 3 movies cost if they sold at the same rate as $25 for 4 movies? Lesson 7 ~ Problem Solving With Proportions 41

11 14. The ratio of boys to girls in a class is 2 : 3. There are 15 girls in the class. How many boys are in the class? 15. The ratio of seventh graders to eighth graders at the dance is 3 : 5. There are 99 seventh graders at the dance. How many eighth graders are at the dance? review determine whether each pair of ratios form a proportion. explain each answer showing cross products _ 4 and _ 5 and and 3 24 Write each fraction as a decimal _ _ _ 4 tic-tac-toe ~ money conversions In December of 2007, $1.00 in U.S. currency equaled: 0.65 Euros - EUR 0.46 Sterling Pounds (United Kingdom) - GBP 0.95 Canadian Dollars - CAD Japanese Yen - JPN 9.46 Mexican New Pesos - MNP Russian Federation Rubles - RFR You can write proportions to find conversions related to currencies. Example: Find the amount of US Dollars needed to buy something worth 30 Euros. Use the ratio $1 to 0.65 Euros. $ Euros = x dollars 30 Euros 0.65x = 30 x = (round to the nearest hundredth) 30 Euros is equal to $ Copy the table. Use proportions to complete it. Find the value of each currency in US Dollars. us dollars 200 eur 200 gbp 200 CAd 200 JPn 200 MnP 200 rfr 2. Research to find today s currency values for $1.00 for each currency listed. Record the values. 3. Copy and complete the chart in Part 1 using the current conversions. 42 Lesson 7 ~ Problem Solving With Proportions

12 similar and congruent Figures Lesson 8 Two figures that have the exact same shape, but not necessarily the exact same size, are called similar figures. The parts of similar figures that match are called corresponding parts. Look at the two triangles below. C D A 4 T O 8 G Corresponding Angles Corresponding sides C and D CT and DG A and O TA and GO T and G AC and OD Corresponding angles in similar figures are congruent or equal in measure. For example, in the triangles above, both C and D measure 35. You can show that two parts of a figure are congruent using the congruence symbol: C D. What do you notice about the corresponding sides CA and DO? AT and OG? CT and DG? They are not equal which means they are not congruent. However, they do all have something in common. Look at the ratio of each of these pairs of corresponding sides. CA DO = _ 3 6 = _ 1 2 AT OG = _ 4 8 = _ 1 2 CT DG = 6 12 = _ 1 2 Each pair of corresponding sides can be reduced to the ratio of 1 : 2. The reduced ratio 1 : 2 is the scale factor for the triangles. This means the sides are proportional. The ratio 1 : 2 means triangle DOG is twice as big as triangle CAT. The two triangles above are similar because the corresponding angles are congruent and the corresponding sides are proportional. You can show that two figures are similar using the similar symbol: CAT ~ DOG. similar not similar Lesson 8 ~ Similar And Congruent Figures 43

13 explore! similar triangles step 1: Draw a right triangle with sides 15 cm and 20 cm on a piece of graph paper. Measure the third side and record its value to the nearest tenth of a centimeter. Label the triangle MAT. 15 cm A M 20 cm T step 2: Draw a right triangle with sides 7.5 cm and 10 cm. Measure the third side to the nearest tenth of a centimeter. Record its value. Label the triangle SIP. S 7.5 cm I 10 cm P step 3: Compare the ratios of the corresponding sides. Write each ratio without decimals in the fraction. MA SI = = = _ 2 1 AT IP = MT SP = step 4: Use a protractor to measure the angles in each triangle. Round your answer to the nearest degree. (m M means measure of angle M.) m M = m S = m A = m I = m T = m P = step 5: Are MAT and SIP similar? Explain your answer. If so, write the scale factor for MAT to SIP. step 6: Draw a third triangle similar to MAT. Explain how you know your triangle is similar to MAT. U example 1 dot is similar to sun. Find the scale factor from dot to sun. O 6 in 6 in 10 in 10 in solution Write a ratio of a side length in DOT to its corresponding side length in SUN. Simplify the ratio. The scale factor from DOT to SUN is 3_ 5 or 3 : 5. D = _ in T S N 5 in 44 Lesson 8 ~ Similar And Congruent Figures

14 example 2 determine whether rectangle CArs is similar to rectangle BIKe. If so, find the scale factor. A 2 cm C 4 cm R S I 3 cm B 6 cm K E solution Determine if corresponding angles are congruent. C B A I R K S E Determine the ratio of corresponding heights. CA BI = _ 2 3 Determine the ratio of corresponding lengths. SC EB = _ 4 6 = _ 2 3 Since the corresponding angles are congruent and the corresponding sides have equal ratios, CARS ~ BIKE. The rectangles have a scale factor of 2 : 3. example 3 Consider the two squares below. Are the two squares similar? explain. 5 m 5 m solution Since every angle is 90, all corresponding angles are equal. Since every side is 5 m, the ratio of any pair of corresponding sides is _ 5 5 = _ 1 1. This means all sides have the same ratio. Yes, they are similar. They have a scale factor of 1 : 1. Although the two squares in example 3 are similar, they have a more specific relationship. When two shapes are the exact same shape and the exact same size, they are congruent figures. Congruent not Congruent Lesson 8 ~ Similar And Congruent Figures 45

15 example 4 a. Is rectangle A similar to rectangle B? b. Is rectangle A similar to rectangle C? rectangle A rectangle B rectangle C 3 cm 7 cm 12 cm 6 cm 4 cm solutions a. All corresponding angles are equal (90 ). 6 cm Look at the ratio of corresponding sides in Rectangle A and Rectangle B. The ratio of short side to short side is 3_ 4. The ratio of long side to long side is 6_ 7. 3_ 4 6_ 7. The sides are not proportional. Rectangle A is NOT similar to Rectangle B. b. All corresponding angles are equal (90 ). Look at the ratio of corresponding sides in Rectangle A and Rectangle C. The ratio of short side to short side is 3_ 6 = 1_ 2. The ratio of long side to long side is 6. The ratios are equal. The sides are proportional. 12 = 1_ 2 Rectangle A is similar to Rectangle C. They have a scale factor of 1 : 2. exercises Find the corresponding sides and corresponding angles to the ones given for each pair of figures. M M T T A N O P 18 9 A R 4 40 O P MN corresponds to Μ AN corresponds to Ν MA corresponds to Α TA corresponds to AR corresponds to TR corresponds to Τ Α R 46 Lesson 8 ~ Similar And Congruent Figures

16 3. C 8 A 4. 4 F 2 I P L S I P 8 M E 2 R CP corresponds to C CA corresponds to A AM corresponds to M MP corresponds to P Y 6 A G 6 PL corresponds to LA corresponds to AY corresponds to PY corresponds to N P L A Y 5. Sketch two similar shapes to the right triangle below, one larger and one smaller. Include angle and side lengths in your drawing. 6. Use this rectangle: 1 in 3 in a. Sketch a rectangle that is congruent to the given rectangle. Label the measurements of the sides. b. Sketch a rectangle that is similar to the given rectangle. Label the measurements of the sides. c. Sketch a rectangle that is not similar or congruent to the given rectangle. Label the measurements of the sides. 7. Are all squares similar? Explain. 8. Are all rectangles similar? Explain. 4 m 45 4 m 5.7 m determine the scale factor for each pair of similar figures Lesson 8 ~ Similar And Congruent Figures 47

17 Determine which pairs of triangles are similar. Explain why they are similar and give their scale factor. triangle A triangle B triangle C triangle d triangle e review solve each proportion _ x 6 = _ x 8 = = a _ 3 5 = 18 y 18. _ 4 6 = _ y _ 5 = x _ 8 Find the perimeter and area of each figure in 4 in 6 m 8 m 10 m 48 Lesson 8 ~ Similar And Congruent Figures

18 tic-tac-toe ~ cookies Chocolate Chip Cookies makes 8 dozen 1 1_ 2 cups sugar 4 1_ 2 cups flour 1 1_ 2 cups brown sugar 2 teaspoons baking soda 2 cups butter 1_ 2 teaspoon salt 4 eggs 4 cups chocolate chips 2 teaspoons vanilla To find the amount of each ingredient needed to make 6 dozen cookies, use the ratio 6_ 8. Write a proportion to find the amount of sugar needed. 6_ 8 = x 1 1_ 8x = 9 x = or 1 1_ 8 cup 2 Use proportions to complete the chart for all nine ingredients. Show all work neatly on a separate piece of paper. Write all decimals as fractions or mixed numbers. Ingredient 4 dozen 6 dozen 10 dozen sugar Brown sugar Butter tic-tac-toe ~ Picture sizes Most picture packages give you the option of buying pictures in five different sizes: 3 in by 5 in 4 in by 6 in 5 in by 7 in 8 in by 10 in 11 in by 14 in 1. A customer picked up her pictures and noticed that the 3 in by 5 in photo cut off a portion of her son s head, but the 8 in by 10 in photo did not. Explain how this happened using ideas about similar figures. 2. Are any of the picture sizes similar to one another? Explain why or why not using diagrams, proportions and the definition of similar figures. 3. Give two different-sized pictures that would be similar to a 4 in by 6 in picture. 4. Find or create a photo or drawing of yourself in two of the sizes listed above. What differences do you notice? Explain the differences using ideas of similar figures. Include the photos or drawings in your work. Lesson 8 ~ Similar And Congruent Figures 49

19 PrOPOrtiOns and similar Figures Lesson 9 Steven stood by a tree in his yard. He wanted to figure out the height of the tree. He noticed that his shadow on the ground was 4 feet and the shadow of the tree was 10 feet. Steven is 6 feet tall. He drew the picture below. 6 ft 10 ft 4 ft Steven s friend, Brandon, came over and looked at his picture. Brandon noticed that Steven and the tree both made right angles with the ground. He noticed that the angle to the sun from the tips of the shadows had to be equal which meant the remaining angles of the triangle were equal. When corresponding angles in two triangles are congruent, the triangles are similar. I know how to figure out how tall the tree is! Brandon cried out. How? asked Steven. example 1 solution Find the height of the tree in steven s yard from the situation above. Since the triangles are similar, the ratios of their corresponding sides are equal. Write a proportion using these ratios to find the height of the tree. Set up the proportion. Let x = the height of the tree. length of tree shadow length of Steven s shadow = height of tree height of Steven 10 4 = _ x 6 Use cross products to solve. 4x = 60 x = 15 The tree is 15 feet tall. 50 Lesson 9 ~ Proportions And Similar Figures

20 example 2 The parallelograms are similar. Find y. 6 in 9 in solution example 3 8 in y in Write a proportion using corresponding sides. 6_ 9 = _ 8 y Use cross products to solve. 6y = 72 y = 12 The missing length is 12 inches. The similar triangles have a scale factor 3 : 5. Find a. a yd 8 yd solution The scale factor is the simplified ratio for all pairs of corresponding sides. Use the scale factor as one of the ratios and the corresponding sides as the other ratio. 3_ 5 = a _ 8 Use cross products to solve. 5a = 24 a = 4.8 The missing length is 4.8 yards. If it is not a sunny day, you can use a mirror to estimate the height of an object. Place a mirror on a flat spot of ground. Back up until you can see the top of the object you are measuring in the mirror. This line of sight creates two similar triangles. If you know your own height to your eyes, the distance you are from the mirror and the distance the object is from the mirror, you can find the height of the object. example 4 shannon wanted to measure the height of her house to the tip of its chimney using a mirror. use the diagram below to find the height of shannon s house. x ft 30 ft 6 ft 5 ft solution height of house height of Shannon s eyes = distance from mirror to house distance from mirror to Shannon Set up the proportion using corresponding sides. x_ 5 = 30 6 Use cross products to solve. 6x = 150 x = 25 Shannon s house is 25 feet high. Lesson 9 ~ Proportions And Similar Figures 51

21 exercises determine the scale factor for each pair of similar figures use proportions to solve for each variable in each pair of similar figures x 5 x x x x 2 y x m Pedro wanted to know the height of the school gym. He measured the shadow from the gym on the ground to be 40 feet and measured his own shadow to be 10 feet. Pedro is 5.5 feet tall. Find the height of the gym. x ft 5.5 ft 40 ft 10 ft 52 Lesson 9 ~ Proportions And Similar Figures

22 14. Michelle used a mirror to find the height of the lamppost in front of her house. She found her eye height to be 6 feet and the distance to the mirror to be 8 feet. The mirror was 20 feet from the lamppost. Find the height of the lamppost. xft 6ft 20ft 8ft 15. Chantel wanted to know the height of the Ferris wheel at the Rose Festival. On a sunny day, she measured the shadow of the Ferris wheel to be 16 feet. The shadow of a 2-foot stick she held up from the ground was 0.8 feet. Find the height of the Ferris wheel. xft 2ft 0.8ft 16ft review Find each unit rate miles 8 gallons laps 16 minutes Complete the conversion feet = inches miles = feet hours = minutes hour = seconds tic-tac-toe ~ how tall? Read through Lesson 9 and look at the examples. Choose two tall objects around you (your home, a tree, a lamppost, the school, ). Use one of the methods given in the lesson to find its height. Method 1: Use shadow lengths along the ground. Be sure to measure your height for the ratio. Method 2: Use a mirror on the ground. Measure your height to your eyes. For each object measured: Draw a picture with the similar triangles labeled. Write a proportion to solve for the height. Solve for the height. Lesson 9 ~ Proportions And Similar Figures 53

23 special ratios FOr similar Figures Lesson 10 explore! Perimeter and area The similar squares below have a scale factor of 2 : 3. The similar triangles have a scale factor of 1 : 4. square A square B triangle A triangle B 2 m 3 m 3 ft 5 ft 12 ft 20 ft step 1: Copy and complete the table. 4 ft 16 ft Perimeter Area Square A Square B Triangle A Triangle B step 2: Find the ratio of the perimeter of Square A to the perimeter of Square B. step 3: How does the ratio of the perimeters of the squares compare to the scale factor of the squares? step 4: Find the ratio of the perimeter of Triangle A to the perimeter of Triangle B. step 5: How does the ratio of the perimeters of the triangles compare to the scale factor of the triangles? step 6: Find the ratio of the area of Square A to the area of Square B. step 7: How does the ratio of the areas of the squares compare to the scale factor of the squares? step 8: Find the ratio of the area of Triangle A to the area of Triangle B. step 9: How does the ratio of the areas of the triangles compare to the scale factor of the triangles? step 10: Suppose there are two similar rectangles with a scale factor of 3 : 5. Predict the ratio of the perimeters and the ratio of the areas. 54 Lesson 10 ~ Special Ratios For Similar Figures

24 example 1 The two rectangles are similar. a. Find their scale factor. b. Find the ratio of their perimeters. c. Find the ratio of their areas. 4 cm 6 cm 6 cm 9 cm solutions a. Choose one pair of corresponding sides and simplify the ratio. The scale factor is 2 : 3. 6_ 9 = _ 2 3 b. The ratio of the perimeter is the same 2 : 3 as the scale factor. Perimeters are 20 cm and 30 cm; 20 : 30 simplifies to 2 : 3. c. The ratio of the areas is the scale 2 2 : 3 2 or 4 : 9 factor squared. Areas are 24 cm 2 and 54 cm 2. The ratio 24 : 54 simplifies to 4 : 9. example 2 The similar figures below have a scale factor of 3 : 4. The smaller figure has a perimeter of 15 in. Find the perimeter of the larger figure. solution Use ratios to write a proportion. First, find the ratio of the perimeters. The ratio of the perimeters 3 : 4 is the same as the scale factor. Write a proportion with x as the perimeter of the larger figure. 3_ 4 = 15 x Use cross products to solve. 3x = 60 x = 20 The larger figure has a perimeter of 20 inches. Lesson 10 ~ Special Ratios For Similar Figures 55

25 example 3 The similar octagons have a scale factor of 2 : 5. a. Find the ratio of their areas. b. The larger figure has an area of 100 ft 2. Find the area of the smaller figure. solutions a. The octagons have a scale factor of 2 : 5. The ratio of the areas is 2² : 5² 4 : 25. b. The ratios of the areas must simplify to 4 : 25. Write a proportion. Let x represent the area of the small octagon = x 100 Use cross products to solve. 25x = 400 x = 16 The smaller octagon has an area of 16 ft ². example 4 solutions two similarly-shaped paintings have perimeters of 20 in and 40 in. a. Find the scale factor. b. Find the ratio of the areas. c. The smaller area is 24 in 2. Find the larger area. a. The scale factor is the same as the ratio of the perimeters. The scale factor is 1 : = 1 _ 2 or 1 : 2 b. Write the ratio of the areas. 1² : 2² 1 : 4. c. Write a proportion with the ratio of the areas. 1_ 4 = 24 x Use cross products to solve. x = 96 The larger area is 96 in². 56 Lesson 10 ~ Special Ratios For Similar Figures

26 exercises determine the scale factor for each pair of similar figures yd 6 yd 4 yd 12 yd 10 ft 10 ft 8 ft 8 ft 5 ft 4 ft in 18 in 3 cm 7 cm 15 in 18 in determine the ratio of the perimeters for each pair of similar figures yd 6 yd 4 yd 12 yd 10 ft 10 ft 8 ft 8 ft 5 ft 4 ft in 18 in 3 cm 7 cm 15 in 18 in determine the ratio of the areas for each pair of similar figures yd 6 yd 4 yd 12 yd 10 ft 10 ft 8 ft 8 ft in 5 ft 18 in 4 ft 3 cm 7 cm 15 in 18 in 13. Two similar right triangles have a scale factor of 1 : 3. a. Find the ratio of their perimeters. b. If the smaller right triangle has a perimeter of 21 m, find the perimeter of the larger right triangle. 14. Two similar rectangles have perimeters of 16 ft and 24 ft. a. Find their scale factor. b. Find the ratio of their areas. Lesson 10 ~ Special Ratios For Similar Figures 57

27 15. Two similar octagons have corresponding sides of 50 in and 20 in. a. Find their scale factor. b. Find the ratio of their perimeters. c. Find the ratio of their areas. d. If the larger shape has an area of 100 in 2, find the area of the smaller octagon. 16. Two similar triangles have perimeters of 16 cm and 4 cm. a. Find their scale factor. b. Find the ratio of their areas. c. If the smaller triangle has an area of 9 cm 2, find the area of the larger triangle. 17. Two similar squares have a scale factor of 3 : 8. a. Find the ratio of their areas. b. The larger square has an area of 128 cm 2. Find the area of the smaller square. review 18. Tricia ran 14 miles in 2 hours. How long will it take her to run 21 miles at this rate? 19. Tiger bowled one game every 15 minutes. At this rate, how many games can Tiger bowl in one and one-half hours? Complete each conversion ,680 feet = miles inches = feet 22. Joy ran 4 miles in 40 minutes. How long will it take her to run 7 miles at this rate? 23. The ratio of dogs to cats at the veterinary clinic is 3 : 4. There are 9 dogs at the veterinary clinic. How many cats are at the veterinary clinic? 24. Sketch a rectangle similar to the one below. Explain why the two rectangles are similar. 18 ft 3 ft 58 Lesson 10 ~ Special Ratios For Similar Figures

28 tic-tac-toe ~ gifts Two similar gifts need to be wrapped. The boxes are shown below. 1. Find the scale factor of the boxes. 8 in 3 in 4 in 1ft 4 in 8 in 6 in 2. A ribbon is going to be put around the middle of each box. a. Find the length of ribbon needed around the smaller box. b. Find the length of ribbon needed around the larger box. c. Find the ratio of the ribbon on the smaller box to the ribbon on the larger box. d. Explain the ratio in part c using ideas about similar figures. 3. Wrapping paper is going to be put around the present. Assume it fits perfectly around the box without any paper overlapping itself. a. Find the area of wrapping paper needed to cover the small box. b. Find the area of wrapping paper needed to cover the large box. c. Find the ratio of wrapping paper needed for the small box to the large box. d. Explain the ratio in part c using ideas about similar figures. 4. Another similar box needed to be wrapped. It needs a ribbon that is three times as large as the small box. a. Find the scale factor of the small box to the new box. b. Find the ratio of ribbon on the small box to the new box. c. Use proportions to find the length of ribbon needed on the new box. d. Find the ratio of wrapping paper on the small box to the new box. e. Use proportions to find the amount of wrapping paper needed on the new box. tic-tac-toe ~ children s story Create a children s book that incorporates the concepts of proportions. At least two different proportions must be created and solved in the story. Your book should have a cover, illustrations and a story line that is appropriate for children. Lesson 10 ~ Special Ratios For Similar Figures 59

29 scale drawings Lesson 11 Janette has a rectangular garden that is 6 feet by 10 feet. She is deciding where to plant her vegetables. She sketches a picture of her garden on paper. Five inches represents the long side of the garden on her paper. What would be the length of the short side of her sketch? Actual Garden 6 ft Sketch of Garden x in 5 in 10 ft Method 1 use ratios with same units Convert all units to inches. 6 ft = 72 in 10 ft = 120 in short side of actual garden = long side of actual garden short side on sketch long side on sketch Write a proportion. 72 x = Use cross products to solve. 120x = 360 x = 3 The short side of the garden on the sketch should be 3 inches. Method 2 use ratios with Mixed units short side of actual garden = long side of actual garden short side on sketch long side on sketch Write a proportion. Make sure the placement of units match in each ratio. 6 ft x in 10 ft = 5 in Use cross products to solve. 10x = 30 x = 3 The short side of the garden on the sketch should be 3 inches. Scale drawings are used in a variety of ways. Architects make blueprints of buildings to help determine the layout of the building as well as the cost to construct the building. You can estimate the length of a trip by using the scale on a map. Some people enjoy making models of cars and airplanes which look exactly like the actual car or airplane they are imitating. Drawing to scale is like looking at similar figures. All corresponding angles must be equal and the measures of all corresponding lengths must be proportional. 60 Lesson 11 ~ Scale Drawings

30 example 1 solutions Alan s map has a scale of 2 inches : 5 miles. a. The distance on the map to Mt. hood Meadows from Alan s home is 16 inches. What is the actual distance from Alan s home to Mt. hood Meadows? b. The actual distance from Alan s home to Portland is 25 miles. Find the distance on the map from his home to Portland. a. Write a proportion using the scale of the map as one ratio. 2 in 5 miles map distance to Mt. Hood Meadows = actual distance to Mt. Hood Meadows 2 in 5 miles = 16 in x miles Use cross products to solve. 2x = 80 x = 40 The actual distance is 40 miles. b. Write a proportion using the scale of the map as one ratio. 2 in 5 miles map distance to Portland = actual distance to Portland 2 in 5 miles = x in 25 miles Use cross products to solve. 5x = 50 x = 10 The distance to Portland on the map is 10 inches. example 2 solution A model car is made with a scale factor of 1 inch : 11 inches or 1 : 11. The actual car is 7 feet 4 inches long. how long is the model car? The scale is in inches, so convert 7 feet 4 inches to inches. 7 ft 4 in = = = 88 in Write a proportion using the scale of 1 : 11 as a ratio length of model car = length of actual car 1 11 = x 88 Use cross products to solve. 11x = 88 x = 8 The model car is 8 inches long. Lesson 11 ~ Scale Drawings 61

31 example 3 Amy is remodeling her kitchen. she is adding two walls to create a breakfast nook. The blueprint shows the dimensions of one wall as 2.5 inches by 4 inches. The other wall is 5 inches by 4 inches on the blueprint. The scale of the blueprint is 1 inch : 2 feet. Find the actual dimensions of the two walls. solution Write a proportion using the scale of 1 in : 2 ft as one of the ratios. 1 in 2 ft = blueprint length of wall actual length of wall Width of first wall: Length of first wall: 1 in 2 ft = 2.5 in xft 1 in 2 ft = 4 in xft x = 5 x = 8 Width of second wall: Length of second wall: 1 in 2 ft = 5 in xft 1 in 2 ft = 4 in xft One wall is 5 feet by 8 feet. The other wall is 10 feet by 8 feet. x = 10 x = 8 exercises A map has a scale of 2 inches : 7 miles. use the given distance on the map to find the actual distance in 2. 7 in 3. 1 ft 4. 1 ft 6 in 5. Lance wanted to make a model of a jet and needed to determine the length of his model. He wanted his model to have a scale of 1 inch : 4 feet. The actual jet is 36 feet 6 inches long. Lance s work to determine the length of his model jet is below. There is an error in his work. Identify the error and solve the problem correctly. 6. A blueprint for a single story home has a scale of 3 inches : 5 feet. a. Find the length of the living room if it is 18 inches on the blueprint. b. Find the width of the kitchen on the blueprint if the actual kitchen is going to be 15 feet wide. c. Find the area of the actual kitchen floor if the length of the kitchen is 20 feet. The width is 15 feet. d. Mario picked out a tile floor for the kitchen that costs $5.50 per square foot. Find the cost of the flooring. 62 Lesson 11 ~ Scale Drawings

32 7. The cities of Portland and Klamath Falls are 270 miles apart. a. One Oregon map has a scale of 1 inch : 24 miles. How far apart are the two cities on the map? b. On another map, the cities are only 3 inches apart. What is the scale of this map? 8. Every 2 inches on the floor plan to the right represents 10 feet in real life. a. Find the scale of the drawing in simplest form. b. Find the actual dimensions of the kitchen. c. Find the actual width of the doorway between the kitchen and family room. d. Can a sofa 6 feet long and 2.5 feet wide fit in the family room? 9. Pedro made a landscape design for his friend s backyard. The yard is 66 feet by 86 feet. He used a scale of 1 inch : 8 feet. Can he sketch the design on a piece of notebook paper 8.5 inches by 11 inches? Explain your answer in 1 2 in Kitchen Family room 1_ 2 in 7 in 10. Bryan enlarges his 3 inch by 5 inch picture of Mt. Hood to make a poster. He used 40 inches as the length of the longest side. What is the length of the shorter side of his poster? 4 in 11. There are two maps of Oregon in a fourth grade classroom that show the Oregon Trail. One map has a scale of 2 inches : 100 miles and the other map has a scale of 3 inches : 135 miles. Both maps show the same areas. Which one has more detail? 12. Estimate the length of the longest distance across Devil s Lake using the scale given. 1.5 miles 13. A model train has a scale of a. The model of an engine is 3 inches long. How long is the actual engine in inches? b. How long is the actual engine in feet? 14. A model of a statue has a scale of 1_ 6. a. The model is 16 inches tall. How tall is the actual statue in inches? b. Write the height of the actual statue in feet. 15. Sarai s mom asked her to clean her room. She decides to also rearrange the furniture in her room. She makes a scale drawing of the room and sketches where the furniture might go. Sarai uses a scale of 1 inch : 2 feet. a. One wall measures 12 feet. How long will the wall be in the drawing? b. The dresser measures 3 feet long. How long will it be in the drawing? c. The bed is 3.25 inches long in the drawing. How long is the bed? Lesson 11 ~ Scale Drawings 63

33 review solve each proportion. 16. x 20 = a = _ 4 = x The directions on a package of strawberry lemonade say to use 8 tablespoons of mix to make 2 quarts. Roy makes 9 quarts of strawberry lemonade. How many tablespoons of mix did he use? 20. Yummy-O cereal is on sale at the store. You can buy a 10 ounce box for $2.50 or a 14 ounce box for $3.36. Use unit rates to determine which one costs less per ounce. 21. Two similar rectangles have a scale factor of 2 : 15. The length of the smaller rectangle is 7 centimeters. Find the length of the larger rectangle. tic-tac-toe ~ home Floor PlAn Use the scale 1 in : 2.5 ft to create a scale drawing of a home. The floor plan must include a minimum of: 1 family room 1 kitchen 1 bathroom 2 bedrooms step 1: Sketch the design of the floor plan. step 2: Draw the floor plan using a ruler. Carefully measure each room. step 3: Next to each wall in the home, label the length of the measurement in the drawing. step 4: Next to each wall in the home, label the length of the actual measurement in the home. step 5: Show the proportions used to solve at least two of the measurements in the home. 64 Lesson 11 ~ Scale Drawings

34 review BLoCK 2 vocabulary congruent corresponding parts scale factor congruent figures proportion similar figures Lesson 6 ~ Write and Solve Proportions solve each proportion. 1. _ 2 5 = x _ 8 = 2 _ x = x _ 3 = 9 _ x = _ 3 y 6. _ a 2 = 5 10 determine whether each pair of ratios forms a proportion _ 3 10 and _ 4 and _ 3 6 and 9 18 Lesson 7 ~ Problem Solving with Proportions 10. Mariah bought three packages of brownies for $9.99. How much would 8 packages of brownies cost? 11. Cody ran a half-marathon (13 miles) in 2 hours. How far would he run in 6 hours at this rate? 12. Ray raked leaves for 3 hours. He filled 2 garbage bags of leaves in this time. How long will it take him to rake 3 garbage bags of leaves at this rate? Block 2 ~ Review 65

35 13. Sung Lu ran 2 laps around the school in 6 minutes. Her brother ran 5 laps. He ran at the same rate as Sung Lu. How long did it take him to run the 5 laps? 14. Reagan bought 20 records at a garage sale for $ Taylor bought 30 records at a store for $24. Which person paid less per record? 15. A store is having a sale on CDs. You can purchase 5 for $25.00 or 9 for $ a. Find the cost per CD for each option. b. Which option is the better deal? c. How much would 5 CDs cost if they sold for the same rate as $36.00 for 9 CDs? Lesson 8 ~ Similar and Congruent Figures 16. Are all squares similar or congruent figures? Explain. For each pair of figures below, find the corresponding sides and corresponding angles to the ones identified. C 17. P O W I 5 R O 3 G D 4 A B I 9 E 12 K CO corresponds to C OW corresponds to O WC corresponds to W RO corresponds to OA corresponds to AD corresponds to DR corresponds to R O A D determine the scale factor for each pair of similar figures Draw a triangle similar to the one given. Label all sides and angles with lengths and degree measures Block 2 ~ Review

36 24. Use the rectangle: 5 ft 2 ft a. Sketch a rectangle that is congruent to the given rectangle. Label the measurements of the sides. b. Sketch a rectangle that is similar to the given rectangle. Label the measurements of the sides. c. Sketch a rectangle that is not similar or congruent to the given rectangle. Label the measurements of the sides. Lesson 9 ~ Proportions and Similar Figures The shapes below are similar. use proportions to solve for each variable x y x x 29. Juan wanted to know the height of a cliff by his school. At 10:00 a.m. he measured the shadow of the cliff along the ground to be 35 feet long. At the same time, he measured his own shadow to be 2 feet long. Juan is 5.5 feet tall. Find the height of the cliff. xft 35 ft 5.5 ft 2 ft 30. Javier used a mirror to find the height of a tree in front of his house. He found his eye height to be 6 feet and the distance to the mirror to be 4 feet. The mirror was 16 feet from the base of the tree. Find the height of the tree. x ft 16 ft 4 ft 6 ft Block 2 ~ Review 67

37 Lesson 10 ~ Special Ratios for Similar Figures For each pair of similar figures: a. Find the scale factor. b. Find the ratio of the perimeters. c. Find the ratio of the areas yd 4 yd 6 ft 9 ft 12 ft 8 ft 4 ft 6 ft m 24 m 18 m 30 m 35. Two similar rectangles have corresponding sides of 8 feet and 12 feet. a. Find the ratio of their perimeters. b. If the smaller rectangle has a perimeter of 36 feet, find the perimeter of the larger rectangle. 36. Two similar octagons have perimeters of 40 inches and 48 inches. a. Find their scale factor. b. If one side of the larger octagon is 12 inches, find the length of the corresponding side on the smaller octagon. 37. Two similar rectangles have perimeters of 15 in and 25 in. a. Find their scale factor. b. Find the ratio of their perimeters. c. Find the ratio of their areas. d. The smaller rectangle has an area of 9 in². Find the area of the larger rectangle. 38. Two similar triangles have a scale factor of 1 : 2. a. Find the ratio of their areas. b. The larger triangle has an area of 24 in². Find the area of the smaller triangle. 68 Block 2 ~ Review

38 Lesson 11 ~ Scale Drawings 39. A map has a scale of 2 inches : 15 miles. Two cities on the map are 6 inches apart. Find the actual distance between the two cities. 40. A map has a scale of 2 inches : 15 miles. A sign next to I-5 says the next rest stop is 20 miles away. How far away is the next rest stop on the map from the location of the sign? 41. A blueprint of a new home has scale 3 inches : 8 feet. a. Find the actual length of the family room if it is 6 inches on the blueprint. b. If one bedroom is 12 feet long, find its length on the blueprint. 42. Patricia has a beautiful 4 inch by 6 inch picture from her latest trip to Crater Lake. She made it into a similarly-shaped poster. The shorter side of the poster measured 3 feet. What was the length of the larger side of the poster? 43. A model car has a scale of a. The actual car is 9 feet long. How long is the model in feet? b. Write the length of the model using inches. 44. Tran made a circular patio in his back yard. The drawing of the patio has a scale of 2 inches : 3 feet. Find the diameter of the actual patio if it is 10 inches on the scale drawing. 10 in x tic-tac-toe ~ contractor The Petersen family remodeled their home. They made their family room larger. They tore down the existing walls and created new walls that were twice as long and twice as tall. The contractor they hired just finished putting drywall on the new walls. He needed to buy baseboards and paint to finish the project. The contractor built the Petersen s home ten years ago. He knows he used 50 feet of baseboards and 1 gallon of paint when the family room was originally built. Since the new walls were doubled in length and height, he bought 100 feet of baseboards and 2 gallons of paint. He quickly realized when he got back to the project that he had enough baseboards, but not enough paint. Write at least two paragraphs explaining why the contractor had enough baseboards, but not enough paint. Use diagrams and similar figures to support your explanation. Block 2 ~ Review 69

39 mick FireFighter/PArAmedic PortlAnd, oregon CAreer FoCus I am a firefighter and paramedic. I work a 24-hour shift, and then have 48 hours off. I respond to emergencies at any time of the day or night. I may be called to fires in houses or buildings. I might also be called for a medical emergency, car accident or any other situation where people need help. A typical day at the fire station includes training or practicing drills related to firefighting. Firefighters also shop and cook for themselves. I help make sure all of our equipment is clean and in good working condition. Most importantly, I need to be prepared for an emergency at all times. I use math whenever I fight a fire. I have to figure out how much pressure is needed in the hoses to ensure water is available everywhere it is needed. I also need to know the pressure of water coming out of a fire hydrant so I can estimate how much water is available. My math must be accurate to ensure that every situation is as safe as possible and that fires are fought most efficiently. I also have to use calculations about water pressures and anchoring on the fireboat. Sometimes, in medical emergencies, I give people medications. I need to use math to figure out how much the patient weighs before I can give a medication. Estimating how much somebody weighs helps me know how much medicine or fluid to give them. If my estimate is off, it could lead to a dangerous situation for the patient. Firefighters must go to college for two years. You must go to college for another two years to become a paramedic. After you are hired as a firefighter and paramedic, you have about 9-12 months of training to complete. A starting firefighter makes about $45,000 per year. An officer makes over $100,000 per year. The best thing about my job is helping people and putting smiles on their faces. When people call 911, they need help and are glad to see me arrive. 70 Block 2 ~ Review