7-1 Ratio and Proportion Warm Up Lesson Presentation Lesson Quiz Holt Geometry

Find the slope of the line through each pair of points. 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

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

Vocabulary ratio proportion cross products

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.

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, a:b, or , where b ≠ 0. For example, the ratios 1 to 2, 1:2, and all represent the same comparison. In a ratio, the denominator of the fraction cannot be zero because division by zero is undefined. Remember!

Examples: In Mr. Alexander’s Geometry class, there are 12 boys and 16 girls. Write the following ratios. 12 12 16 16 16 1. Boys to Girls 2. Girls to Students 3. Girls to Boys 28

In the United States House of Representatives there are 435 seats In the United States House of Representatives there are 435 seats. Of those, 70 are occupied by women. Write the ratio of men to women in the US House of Representatives. 365 70

Solving problems with ratios! To simplify a ratio, we divide out the common factor. So, when we solve we are looking for that factor. How do we put the 4 back in? Divide 4 out So to make ratios big again we multiply!

The ratio of the lengths of an isosceles triangle is 4:4:7, and its perimeter is 52.5 cm. What are the lengths of the sides of the triangle? 4 4 7 We know we have a missing factor. What do we call it? How do we find perimeter? x x x 15x = 52.5 4(3.5) = 14 7(3.5) = 24.5 X = 3.5

Rate of change Rise Run m One common ratio is slope, which is the comparison of the change in y to the change in x. This can also be expressed as Rate of change Rise Run m

Example 7 Write a ratio expressing the slope of the line. 6 4

Example 8 Write the slopes as a ratio for points A(7, 9) and B(2, -6). -6 – 9 2 – 7 Substitute the given values. -15 -5 3 1 = Simplify.

A proportion is an equation stating that two ratios are equal. When you cross multiply you create equal cross products. For example in the proportion , ad = bc. Once you have cross multiplied you need to solve for the variable using algebra.

Example 9: Solving Proportions Solve the proportion. 4(65) = k(10) Cross Products Property 260 = 10k Simplify. k = 26 Divide both sides by10.

Example 10: Solving Proportions Solve the proportion. p(p)= 4(9) Cross Products Property p2 = 36 Simplify. p = 6 Find the square root of both sides.

Example 11 Solve the proportion. 3(x + 8) = 4(x + 3) Cross Products Property 3x + 24 = 4x + 12 Distribute. 24 = x + 12 Subtract both sides by 3x. 12 = x Subtract both sides by 12.

Example 12 Given that , complete the following equations. Cross Products are 7a and 5b 7a = 5b

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

Understand the Problem Example 13: Problem-Solving Application The scale of a map of downtown Dallas is 1.5 cm:300 m. If the distance between Union Station and the Dallas Public Library is 6 cm, what is the actual distance? 1 Understand the Problem The answer will be the distance from the Union Station to the Dallas Public Library.

Example 13 Continued 2 Make a Plan Let x be the distance from the Union Station to the Dallas Public Library. Write a proportion that compares the ratios of the width to the length. Distance in cm Distance in m

Example 13 Continued Solve 3 6(300) = x(1.5) Cross Products Property 1800 = 1.5x Simplify. x = 1200 Divide both sides by 1.5. The distance from the Union Station to the Dallas Public Library is 1200 m.

Example 13 Continued 4 Look Back Check the answer in the original problem. The ratio of the scale distance to actual distance is 6:1200, or 1:200. The ratio of the given scale is also 1:200. So the ratios are equal, and the answer is correct. 6/1200 = .005 1.5/300 = .005

Example 14 The $250,000 budget for a local shelter is allocated proportionally to the men’s and women’s departments according to the population in the shelter by gender. If there are 1946 women and 399 men in the shelter, what amount rounded to the nearest dollar is allocated to the men’s department?

Example 15 After an election in a small town, the newspaper reported that 42% of the registered voters actually voted. If 12,000 people voted, how many people are registered to vote in the town?

Example 16 A student wanted to find the height of a statue of a pineapple in Nambour, Australia. She measured the pineapple’s shadow at 8 ft 9in and her own shadow at 2 ft. The student’s height is 5 ft 4 in. What is the height of the pineapple?

Example 17 The Lincoln Memorial in Washington, D.C., is approximately 57 m long and 36 m wide. If you would want to make a scale drawing of the base of the building using a scale of 1 cm: 15 m, what would be the dimensions of the scale drawing?

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? 15°, 75°, 90° 3 7 or –7 18 in.