Ratios and Proportions. To write ratios and solve proportions

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1 7-1 Ratios and Proportions Content Standard Prepares for G.SRT. Use... similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Objective To write ratios and solve proportions The year the team had the most wins is not necessarily the year in which it had the best record. MTHEMTICL PRCTICES The table at the right gives the wins and losses of a baseball team. In which year(s) did the team have the best record? Eplain. In the Solve It, you compared two quantities for four years. Year Wins Losses Essential Understanding You can write a ratio to compare two quantities. Lesson L Vocabulary V ratio proportion etremes means How can you write the heights using the same unit? You can convert the height of the Senator to inches or the height of the bonsai tree to feet. ratio is a comparison of two quantities by division. You can write the ratio of two numbers a and b, where b 0, in three ways: a b, a b, and a to b. You usually epress a and b in the same unit and write the ratio in simplest form. Problem 1 Writing a Ratio onsai Trees The bonsai bald cypress tree is a small version of a full-size tree. Florida bald cypress tree called the Senator stands 118 ft tall. What is the ratio of the height of the bonsai to the height of the Senator? Epress both heights in the same unit. To convert 118 ft 1 in. to inches, multiply by the conversion factor 1 ft. 118 ft 1 in. 118 ft 1 (118 1) in. 11 in. 1 ft Write the ratio as a fraction in simplest form. height of bonsai S height of Senator S 1 in. 118 ft 1 in. ( ) in. 11 in. ( 7) in. 7 The ratio of the height of the bonsai to the height of the Senator is 7 or bonsai tree is 18 in. wide and stands ft tall. What is the ratio of the width of the bonsai to its height? 1 in. Chapter 7 Similarity

2 Problem Dividing a Quantity Into a Given Ratio Fundraising Members of the school band are buying pots of tulips and pots of daffodils to sell at their fundraiser. They plan to buy 10 pots of flowers. The ratio number of tulip pots number of daffodil pots will be. How many pots of each type of flower should they buy? How do you write epressions for the numbers of pots? Multiply the numerator and the denominator by the factor.. number of tulip pots If the ratio number of daffodil pots is, it must be in the form. The total number of flower pots is 10. Use this fact to write an equation. Then solve for. Substitute for in the epressions for the numbers of pots. Write the answer in words. Let the number of tulip pots. Let the number of daffodil pots () 8 () 7 The band members should buy 8 tulip pots and 7 daffodil pots.. The measures of two supplementary angles are in the ratio 1. What are the measures of the angles? n etended ratio compares three (or more) numbers. In the etended ratio a b c, the ratio of the first two numbers is a b, the ratio of the last two numbers is b c, and the ratio of the first and last numbers is a c. How do you use the solution of the equation to answer the question? Substitute the value for in the epression for the length of the longest side. Problem Using an Etended Ratio The lengths of the sides of a triangle are in the etended ratio : :. The perimeter of the triangle is 98 in. What is the length of the longest side? Sketch the triangle. Use the etended ratio to label the sides with epressions for their lengths. 98 The perimeter is 98 in Simplify. 7 Divide each side by 1. The epression that represents the length of the longest side is. (7), so the length of the longest side is in.. The lengths of the sides of a triangle are in the etended ratio 7 9. The perimeter is 0 cm. What are the lengths of the sides? Lesson 7-1

3 Essential Understanding If two ratios are equivalent, you can write an equation stating that the ratios are equal. If the equation contains a variable, you can solve the equation to find the value of the variable. n equation that states that two ratios are equal is called a proportion. The first and last numbers in a proportion are the etremes. The middle two numbers are the means. etremes T T etremes S means S c c means Key Concept Cross Products Property Words In a proportion, the product of the etremes equals the product of the means. Symbols If a b c d, where b 0 and d 0, then ad bc. Eample?? 1 1 Here s Why It Works egin with a b c d, where b 0 and d 0. bd? a b c d? bd Multiply each side of the proportion by bd. bd 1? a b c d? bd 1 Divide the common factors. ad bc Simplify. Does the solution check? 0? 0? Problem Solving a Proportion lgebra What is the solution of each proportion? () Cross Products Property Simplify. Solve for the variable. y 1 9 y (y 1 ) 9y y 1 1 9y 1 y y The solution is or.8. The solution is.. What is the solution of each proportion? a. 9 a 1 1 b. m 1 1 m Chapter 7 Similarity

4 Using the Properties of Equality, you can rewrite proportions in equivalent forms. Key Concept Properties of Proportions a, b, c, and d do not equal zero. Property How to pply It (1) a b c d is equivalent to a b d c. Write the reciprocal of each ratio. becomes. () a b c d is equivalent to a c b. Switch the means. d becomes. () a b c d is equivalent to a b b c d. In each ratio, add the denominator to the d numerator. becomes. Problem Writing Equivalent Proportions How do you decide which property of proportions applies? Look at how the positions of the known parts of the incomplete proportion relate to their positions in the original proportion. In the diagram, y 7. What ratio completes the equivalent proportion y? Justify your answer. Method 1 y 7 y 7 Property of Proportions () The ratio that completes the proportion is 7. Method y 7 7 y 7 y 7y 7y y 7 Cross Products Property To solve for y, divide each side by 7y. Simplify. y 7. For parts (a) and (b), use the proportion y 7. What ratio completes the equivalent proportion? Justify your answer. a. b. y 7 7 c. Reasoning Eplain why y 7 7 is an equivalent proportion to y 7. Lesson 7-1

5 Lesson Check Do you know HOW? 1. To the nearest millimeter, a cell phone is 8 mm long and mm wide. What is the ratio of the width to the length?. Two angle measures are in the ratio 9. Write epressions for the two angle measures in terms of the variable.. What is the solution of the proportion 0 z?. For a 1 7 b complete each equivalent proportion. a. a 7 b. a 7 7 c. 7 a Do you UNDERSTND?. Vocabulary What is the difference between a ratio and a proportion?. Open-Ended The lengths of the sides of a triangle are in the etended ratio 7. What are two possible sets of side lengths, in inches, for the triangle? 7. Error nalysis What is the error in the solution of the proportion shown at the right? 8. What is a proportion that has means and 18 and etremes 9 and 1? MTHEMTICL PRCTICES Practice and Problem-Solving Eercises MTHEMTICL PRCTICES Practice Write the ratio of the first measurement to the second measurement. 9. length of a tennis racket: ft in. 10. height of a table tennis net: in. length of a table tennis paddle: 10 in. height of a tennis net: ft 11. diameter of a table tennis ball: 0 mm 1. length of a tennis court: yd diameter of a tennis ball:.8 cm length of a table tennis table: 9 ft See Problem aseball baseball team played 1 regular season games. The ratio of the number of games they won to the number of games they lost was. How many games did they win? How many games did they lose? 1. The measures of two supplementary angles are in the ratio 7. What is the measure of the larger angle? 1. The lengths of the sides of a triangle are in the etended ratio 7 9. The perimeter of the triangle is 88 cm. What are the lengths of the sides? 1. The measures of the angles of a triangle are in the etended ratio. What is the measure of the largest angle? See Problem. See Problem. lgebra Solve each proportion. See Problem b y y n. n 8 n Chapter 7 Similarity

6 pply In the diagram, a b. Complete each statement. Justify your answer. 7. a b 8. a 9. b a b b. b b Coordinate Geometry Use the graph. Write each ratio in simplest form.. C D. E EC. slope of E. slope of ED 7. Think bout a Plan The area of a rectangle is 10 in.. The ratio of the length to the width is. Find the length and the width. What is the formula for the area of a rectangle? How can you use the given ratio to write epressions for the length and width? a 8 O y See Problem. E C D rt To draw a face, you can sketch the head as an oval and then lightly draw horizontal lines to help locate the eyes, nose, and mouth. You can use the etended ratios shown in the diagrams to help you place the lines for an adult s face or for a baby s face. 8. If E 7 cm in the diagram, find, C, CD, and DE. 9. You draw a baby s head as an oval that is 1 in. from top to bottom. a. How far from the top should you place the line for the eyes? b. Suppose you decide to make the head an adult s head. How far up should you move the line for the eyes? lgebra Solve each proportion y 9y 1. a 1 7 a. 1. b 1 b 1. The ratio of the length to the width of a rectangle is 9. The width of the rectangle is mm. Write and solve a proportion to find the length.. Open-Ended Draw a quadrilateral that satisfies this condition: The measures of the consecutive angles are in the etended ratio 7.. Reasoning The means of a proportion are and 1. List all possible pairs of positive integers that could be the etremes of the proportion. 7. Writing Describe how to use the Cross Products Property to determine whether 10 1 is a true proportion. 8. Reasoning Eplain how to use two different properties of proportions to change the proportion into the proportion. Lesson 7-1 7

7 C Challenge Complete each statement. Justify your answer. 9. If m 9n, then m n. 0. If 0 t 18 r, then t r a. 1. If. If a b c a b, then d c d.. If a b c a c, then d b d. lgebra Use properties of equality to justify each property of proportions.. a b c d is equivalent to b a d c.. a b c d is equivalent to a c b d. 7. a b c d is equivalent to a b b c d d. lgebra Solve each proportion for the variable(s). b, then a.. If a b c a b, then d b y Standardized Test Prep ST/CT 1. In the diagram at the right, D is the perpendicular bisector of C. What is the length of E?. The measures of the angles of a triangle are in the etended ratio 1. What is the measure of the largest angle? E D 8 C. In the diagram at the right, C is an isosceles triangle with base C and mc 0. What is the value of? C E. In PQR, PQ 10 in. and QR 1 in. RP must be less than inches. What is the value of? D. What is the length of the line segment joining P(, ) and Q(9, )? Mied Review. Write a coordinate proof of this statement: The diagonals of a square are perpendicular. In each eercise, identify two statements that contradict each other. See Lesson -9. See Lessons 1- and I. PQR is isosceles. II. PQR is an obtuse triangle. III. PQR is scalene. 8. I. 1 II. 1 and are complementary. III. m1 m 180 Get Ready! To prepare for Lesson 7-, do Eercise C HIJ. Name three pairs of congruent angles and three pairs of congruent sides. See Lesson Chapter 7 Similarity