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Homework: Chapter Page 499 # 1-27 Odds (Must Show Work)

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**Warm Up 2 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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**Objectives Write and simplify ratios.**

Use proportions to solve problems.

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Vocabulary ratio Proportion Extended Proportion cross products

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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**

A ratio compares two numbers by division. The ratio , of two numbers a and b can be written as a to 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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Ratios & Slope Write a ratio expressing the slope of l. Substitute the given values. Simplify.

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TEACH! Ratio & Slope 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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**Example 1 : Similarity statement & Ratios**

Are the triangles similar? If they are, write a similarity statement and give the ratio. If they are not, explain why. 15 20 24 16 12 18 F E D C B A The corresponding angle pairs are congruent. Find the ratios and compare.

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A Golden Rectangle is a rectangle that can be divided into a square and a rectangle that is similar to the original rectangle. The ratio of the length of a Golden Rectangle to its width is called the Golden Ratio. A B C D E F C B F E

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A Golden Rectangle is a rectangle that can be divided into a square and a rectangle that is similar to the original rectangle. The ratio of the length of a Golden Rectangle to its width is called the Golden Ratio. A B C D E F Find the Golden Ratio. 1 1 X-1 Corresponding sides are proportional X Solve Golden Ratio is 1:1.618

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**A ratio can involve more than two numbers**

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 = After like terms are combined, x = So x = 6. The length of the shortest side is x = 4(6) = 24 cm.

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

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**A proportion is an equation stating that two ratios are equal**

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. - An extended proportion is when three or more ratios are equal, such as:

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**In Algebra 1 you learned the Cross Products Property**

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 3: Solving Proportions**

Solve the proportion. 7(72) = x(56) Cross Products Property 504 = 56x Simplify. x = 9 Divide both sides by 56.

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**Example 3: Solving Proportions**

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

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TEACH! Example 3A Solve the proportion. 3(56) = 8(x) Cross Products Property 168 = 8x Simplify. x = 21 Divide both sides by 8.

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TEACH! Example 3B Solve the proportion. d(2) = 3(6) Cross Products Property 2d = 18 Simplify. d = 9 Divide both sides by 2.

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

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

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

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Example 5 Continued 4 Look Back Check the answer in the original problem. The ratio of the width to the length of the actual room is :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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**Understand the Problem**

TEACH! Example 5 Suppose the special-effects team of “Lord of The Ring” made a model with a height of 9.2 ft and a width of 6 ft. What is the height of the tower presented in the movie if the scale proportion is 1:166? 1 Understand the Problem The answer will be the height of the tower.

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**TEACH! 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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**TEACH! Example 5 Continued**

Solve 3 9.2(996) = 6(x) Cross Products Property = 6x Simplify. = x Divide both sides by 6. The height of the actual tower is feet.

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**TEACH! Example 5 Continued**

4 Look Back 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 :996, or 9.2:6. So the ratios are equal, and the answer is correct.

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**4. Given that 14a = 35b, find the ratio of a to b in simplest form.**

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. 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? 15°, 75°, 90° 7 or –7 X = 3 18 in.

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