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

Do Now 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.

Objectives TSW write and simplify ratios. TSW use proportions to solve problems. TSW use ratios to make indirect measurements. TSW use scale drawings to solve problems.

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!

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

Example 2 Given that two points on m are C(–2, 3) and D(6, 5), write a ratio expressing the slope of m.

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.

Example 3: 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?

Example 4 The ratio of the angle measures in a triangle is 1:6:13. What is the measure of each angle?

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.

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.

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

Example 5: Solving Proportions Solve the proportion.

Example 5.5: Solving Proportions Solve the proportion.

Example 5.6 Solve the proportion.

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

Example 6: Using Properties of Proportions Given that 18c = 24d, find the ratio of d to c in simplest form.

Example 7 Given that 16s = 20t, find the ratio t:s in simplest form.

Example 8: Problem-Solving Application 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?

Example 8 During the filming of The Lord of the Rings, the special-effects team built a model of Sauron’s tower with a height of 8 m and a width of 6 m. If the width of the full-size tower is 996 m, what is its height?

Example 9 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?

More Do Now Convert each measurement. 1. 6 ft 3 in. to inches 2. 5 m 38 cm to centimeters Find the perimeter and area of each polygon. 3. square with side length 13 cm 4. rectangle with length 5.8 m and width 2.5 m

Indirect measurement is any method that uses formulas, similar figures, and/or proportions to measure an object. The following example shows one indirect measurement technique.

Whenever dimensions are given in both feet and inches, you must convert them to either feet or inches before doing any calculations. Helpful Hint

Example 10: Measurement Application Tyler wants to find the height of a telephone pole. He measured the pole’s shadow and his own shadow and then made a diagram. What is the height h of the pole?

Example 11 A student who is 5 ft 6 in. tall measured shadows to find the height LM of a flagpole. What is LM?

Check It Out! Example 1 Continued Step 2 Find similar triangles. Because the sun’s rays are parallel, L  G. Therefore ∆JGH ~ ∆NLM by AA ~. Step 3 Find h. Corr. sides are proportional. Substitute 66 for BC, h for LM, 60 for JH, and 170 for MN. Cross Products Prop. 60(h) = 66  170 h = 187 Divide both sides by 60. The height of the flagpole is 187 in., or 15 ft. 7 in.

A scale drawing represents an object as smaller than or larger than its actual size. The drawing’s scale is the ratio of any length in the drawing to the corresponding actual length. For example, on a map with a scale of 1 cm : 1500 m, one centimeter on the map represents 1500 m in actual distance.

A proportion may compare measurements that have different units. Remember!

Example 12: Solving for a Dimension On a Wisconsin road map, Kristin measured a distance of 11 in. from Madison to Wausau. The scale of this map is 1inch:13 miles. What is the actual distance between Madison and Wausau to the nearest mile?

Example 13 Find the actual distance between City Hall and El Centro College.

Check It Out! Example 2 Continued To find the actual distance x write a proportion comparing the map distance to the actual distance. 1x = 3(300) Cross Products Prop. Simplify. x  900 The actual distance is 900 meters, or 0.9 km.

Example 14: Making a Scale Drawing Lady Liberty holds a tablet in her left hand. The tablet is 7.19 m long and 4.14 m wide. If you made a scale drawing using the scale 1 cm:0.75 m, what would be the dimensions to the nearest tenth?

Example 15 The rectangular central chamber of the Lincoln Memorial is 74 ft long and 60 ft wide. Make a scale drawing of the floor of the chamber using a scale of 1 in.:20 ft.

Check It Out! Example 3 Continued Set up proportions to find the length l and width w of the scale drawing. 20w = 60 w = 3 in 3.7 in. 3 in.

Example 16: Using Ratios to Find Perimeters and Areas Given that ∆LMN:∆QRT, find the perimeter P and area A of ∆QRS.

Example 17 ∆ABC ~ ∆DEF, BC = 4 mm, and EF = 12 mm. If P = 42 mm and A = 96 mm2 for ∆DEF, find the perimeter and area of ∆ABC.

Lesson Quiz: Part I 1. Maria is 4 ft 2 in. tall. To find the height of a flagpole, she measured her shadow and the pole’s shadow. What is the height h of the flagpole? 2. A blueprint for Latisha’s bedroom uses a scale of 1 in.:4 ft. Her bedroom on the blueprint is 3 in. long. How long is the actual room? 25 ft 12 ft

Lesson Quiz: Part II 3. ∆ABC ~ ∆DEF. Find the perimeter and area of ∆ABC. P = 27 in., A = 31.5 in2

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.