Similar Shapes and Scale Drawings 4.3 Similar Shapes and Scale Drawings How can you use scale drawings to solve problems?

Texas Essential Knowledge and Skills The student is expected to: 7.5.C Solve mathematical and real-world problems involving similar shape and scale drawings. Mathematical Processes 7.1.A Apply mathematics to problems arising in everyday life, society, and the workplace.

Warm Up Write each fraction in the simplest form. 4 48 1 12 9 135 1 15 2. 1. Convert the following measurements. 3. 192 inches = feet 16 4. 18.5 feet = inches 222 5. 324 inches = feet 27

A scale model is a proportional model of a three-dimensional object A scale model is a proportional model of a three-dimensional object. Its dimensions are related to the dimensions of the actual object by a ratio called the scale factor. This means that each dimension of the model is of the corresponding dimension of the actual train. 1 87

A scale is the ratio between two sets of measurements A scale is the ratio between two sets of measurements. Scales can use the same units or different units. The photograph shows a scale drawing of the model train. A scale drawing is a proportional drawing of an object. Both scale drawings and scale models can be smaller or larger than the objects they represent.

Additional Example 1: Finding a Scale Factor Identify the scale factor. Room Blueprint Length (in.) 144 18 Width (in.) 108 13.5 blueprint length room length 18 144 Write a ratio using one of the dimensions. = 1 8 = Simplify. A scale factor is always the ratio of the model’s dimensions to the actual object’s dimensions. Caution! 1 8 The scale factor is .

Check It Out: Example 1 Identify the scale factor. Model Aircraft Blueprint Length (in.) 12 2 Wing span (in.) 18 3 blueprint length aircraft length 2 12 = Write a ratio using one of the dimensions. 1 6 = Simplify. The scale factor is . This is reasonable because of the length of the model is 3 in. The length of the blueprint is 2, which is less. is less than . 1 6 4

Additional Example 2: Using Scale Factors to Find Unknown Lengths A photograph was enlarged and made into a poster. The poster is 20.5 inches by 36 inches. The scale factor is . Find the size of the photograph. 5 1 poster photo 5 1 Think: = 36 l 5 1 = Write a proportion to find the length l. 5l = 36 Find the cross products. l = 7.2 Divide.

Additional Example 2 Continued A photograph was enlarged and made into a poster. The poster is 20.5 inches by 36 inches. The scale factor is . Find the size of the photograph. 5 1 poster photo 5 1 Think: = 20.5 w 5 1 Write a proportion to find the width w. = 5w = 20.5 Find the cross products. w = 4.1 Divide. The photo is 7.2 in. long and 4.1 in. wide.

Additional Example 3: Measurement Application On a road map, the distance between Pittsburgh and Philadelphia is 7.5 inches. What is the actual distance between the cities if the map scale is 1.5 inches = 60 miles? Let d be the actual distance between the cities. 1.5 60 7.5 d = Write a proportion. 1.5 · d = 60 · 7.5 Find the cross products. 1.5d = 450 Multiply. 1.5d 1.5 450 1.5 = Divide both sides by 1.5. d = 300 The distance between the cities is 300 miles.

Check It Out: Example 3 On a road map, the distance between Dallas and Houston is 7 inches. What is the actual distance between the cities if the map scale is 1 inch = 50 kilometers? Let d be the actual distance between the cities. 1 50 7 d = Write a proportion. 1 · d = 50 · 7 Find the cross products. 1d = 350 Multiply. d = 350 The distance between the cities is 350 kilometers.

ADDITIONAL EXAMPLE 1 Joanne has a scale drawing of her backyard that includes a garden bed that measures 16 inches long and 25 inches wide. What is the area of the actual garden bed? 625 ft2

4.3 LESSON QUIZ 7.5.C 1. A scale drawing of a billboard uses the scale 4 cm:9 ft. The length of the billboard in the drawing is 11 cm. How long is the actual billboard? 24.75 ft 2. A scale drawing of a dance floor is shown. What is the area of the actual dance floor? 788.48 ft2

3. A bookcase measures 13 feet wide and 24 feet tall. What would the bookcase’s measurements be on a scale drawing using the scale 3 cm:2 ft? 19.5 cm wide, 36 cm tall 4. Bob makes a scale drawing of a statue using the scale 1 cm:5 ft. His drawing measures 12 cm. Kia makes a scale drawing of the same statue using the scale 1 cm:4 ft. How many centimeters tall is the statue in Kia’s drawing? 15 cm

A billboard is 2. 5 times as long as it is wide A billboard is 2.5 times as long as it is wide. The area of the billboard is 2,250 ft2. A scale drawing is made of the billboard, and the area of the scale drawing is 160 in2. What is the scale used in the scale drawing? Explain. 4 in:15 feet; Since the length of the billboard is 2.5 times the width, the equation for the area of the billboard is (w)(2.5w) = 2,250 ft2, where w is the width and 2.5w is the length. Using trial and error, students can find that the width is 30 ft and the length is 75 ft. Since the scale drawing is

similar in shape to the billboard, the drawing is also 2 similar in shape to the billboard, the drawing is also 2.5 times as long as it is wide, and (w)(2.5w) = 160 in2. Students can find the width is 8 inches, and the length is 20 inches. Since the drawing and the billboard are similar, 8 inches on the drawing corresponds to 30 feet on the billboard. The scale used is 8 in:30 ft, which can be simplified as 4 in:15 ft.

How can you use scale drawings to solve problems? Sample answer: You use scale drawings to represent measurements of actual objects or places. You can find dimensions of actual objects by making and completing a table, or by writing and solving proportions.