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Geometry Chapter 7 7.1 Ratios and Proportions

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Warm Up Find the slope of the line through each pair of points. 1. (1, 5) and (3, 9) 2. (–6, 4) and (6, –2) Solve each equation. 3. 4x + 5x + 6x = 45 4. (x – 5) 2 = 81 5. Write in simplest form. 2 x = 3 x = 14 or x = –4

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Write and simplify ratios. Use proportions to solve problems. Objectives

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ratio proportion extremes means cross products Vocabulary

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The Lord of the Rings movies transport viewers to the fantasy world of Middle Earth. Many scenes feature vast fortresses, sprawling cities, and bottomless mines. To film these images, the moviemakers used ratios to help them build highly detailed miniature models.

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A ratio compares two numbers by division. The ratio of two numbers a and b can be written as a to b, a:b, or, where b ≠ 0. For example, the ratios 1 to 2, 1:2, and all represent the same comparison.

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In a ratio, the denominator of the fraction cannot be zero because division by zero is undefined. Remember!

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Example 1: Writing Ratios Write a ratio expressing the slope of l. Substitute the given values. Simplify.

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In Your Notes! Example 1 Given that two points on m are C(–2, 3) and D(6, 5), write a ratio expressing the slope of m. Substitute the given values. Simplify.

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A ratio can involve more than two numbers. For the rectangle, the ratio of the side lengths may be written as 3:7:3:7.

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Example 2: Using Ratios The ratio of the side lengths of a triangle is 4:7:5, and its perimeter is 96 cm. What is the length of the shortest side? Let the side lengths be 4x, 7x, and 5x. Then 4x + 7x + 5x = 96. After like terms are combined, 16x = 96. So x = 6. The length of the shortest side is 4x = 4(6) = 24 cm.

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In Your Notes! Example 2 The ratio of the angle measures in a triangle is 1:6:13. What is the measure of each angle? x + y + z = 180° x + 6x + 13x = 180° 20x = 180° x = 9° y = 6x y = 6(9°) y = 54° z = 13x z = 13(9°) z = 117°

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A proportion is an equation stating that two ratios are equal. In the proportion, the values a and d are the extremes. The values b and c are the means. When the proportion is written as a:b = c:d, the extremes are in the first and last positions. The means are in the two middle positions.

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In Algebra 1 you learned the Cross Products Property. The product of the extremes ad and the product of the means bc are called the cross products.

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The Cross Products Property can also be stated as, “In a proportion, the product of the extremes is equal to the product of the means.” Reading Math

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Example 3A: Solving Proportions Solve the proportion. Cross Products Property Simplify. Divide both sides by 56. 7(72) = x(56) 504 = 56x x = 9

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Example 3B: Solving Proportions Solve the proportion. Cross Products Property(z – 4) 2 = 5(20) Simplify.(z – 4) 2 = 100 Find the square root of both sides. (z – 4) = 10 (z – 4) = 10 or (z – 4) = –10 Rewrite as two eqns. z = 14 or z = –6 Add 4 to both sides.

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In Your Notes! Example 3a Solve the proportion. Cross Products Property Simplify. Divide both sides by 8. 3(56) = 8(x) 168 = 8x x = 21

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In Your Notes! Example 3b Solve the proportion. Cross Products Property Simplify. Divide both sides by 8. 2y(4y) = 9(8) 8y 2 = 72 y 2 = 9 Find the square root of both sides. y = 3 Rewrite as two equations.y = 3 or y = –3

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In Your Notes! Example 3c Solve the proportion. Cross Products Property Simplify. Divide both sides by 2. d(2) = 3(6) 2d = 18 d = 9

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In Your Notes! Example 3d Solve the proportion. Cross Products Property(x + 3) 2 = 4(9) Simplify.(x + 3) 2 = 36 Find the square root of both sides. (x + 3) = 6 (x + 3) = 6 or (x + 3) = –6Rewrite as two eqns. x = 3 or x = –9Subtract 3 from both sides.

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The following table shows equivalent forms of the Cross Products Property.

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Example 4: Using Properties of Proportions Given that 18c = 24d, find the ratio of d to c in simplest form. 18c = 24d Divide both sides by 24c. Simplify.

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In Your Notes! Example 4 Given that 16s = 20t, find the ratio t:s in simplest form. 16s = 20t Divide both sides by 20s. Simplify.

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Example 5: Problem-Solving Application 1 Understand the Problem The answer will be the length of the room on the scale drawing. Marta is making a scale drawing of her bedroom. Her rectangular room is 12 feet wide and 15 feet long. On the scale drawing, the width of her room is 5 inches. What is the length?

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Example 5 Continued 2 Make a Plan Let x be the length of the room on the scale drawing. Write a proportion that compares the ratios of the width to the length.

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Solve 3 Example 5 Continued Cross Products Property Simplify. Divide both sides by 12.5. 5(15) = x(12.5) 75 = 12.5x x = 6 The length of the room on the scale drawing is 6 inches.

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Look Back 4 Example 5 Continued Check the answer in the original problem. The ratio of the width to the length of the actual room is 12 :15, or 5:6. The ratio of the width to the length in the scale drawing is also 5:6. So the ratios are equal, and the answer is correct.

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In Your Notes! Example 5 What if...? Suppose the special-effects team made a different model with a height of 9.2 m and a width of 6 m. What is the height of the actual tower? 1 Understand the Problem The answer will be the height of the tower.

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In Your Notes! Example 5 Continued 2 Make a Plan Let x be the height of the tower. Write a proportion that compares the ratios of the height to the width.

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Solve 3 In Your Notes! Example 5 Continued Cross Products Property Simplify. Divide both sides by 6. 9.2(996) = 6(x) 9163.2 = 6x 1527.2 = x The height of the actual tower is 1527.2 feet.

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Look Back 4 Check the answer in the original problem. The ratio of the height to the width of the model is 9.2:6. The ratio of the height to the width of the tower is 1527.2:996, or 9.2:6. So the ratios are equal, and the answer is correct. In Your Notes! Example 5 Continued

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Lesson Quiz 1. The ratio of the angle measures in a triangle is 1:5:6. What is the measure of each angle? Solve each proportion. 2. 3. 4. Given that 14a = 35b, find the ratio of a to b in simplest form. 5. An apartment building is 90 ft tall and 55 ft wide. If a scale model of this building is 11 in. wide, how tall is the scale model of the building?

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