A scale factor can be used to generate a dilation of a figure. If the scale factor is less than 1, then the image, or new figure, will be smaller than the preimage, or original figure. This type of dilation is called a reduction.

An art studio is promoting three-dimensional chalk art. They are sending out postcards with a three-dimensional art image. The picture on the post card has to be 3 inches in width by 4 inches in length. The original image is 5 inches in width by b inches in length. What is the length of the original image?

Source: 3D chalk drawings 3, wstera2, Flickr

Drag the tiles representing the numbers to their appropriate place in the proportion below. Use the proportion to solve the problem. Use your results to answer the questions that follow.

• What is the length of the original picture?

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  62 over 3 2 3 inches

• What scale factor was applied to the original picture to create the postcard picture?

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  5 over 3 5 3

Pause and Reflect

In the example problem, you used information provided to determine a scale factor and then determine the missing length. How could you use a proportion to obtain an answer?

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In similar figures, the ratios of the lengths of corresponding sides are equal. Write an equation where the ratios of corresponding side lengths are set equal to each other. Then solve the equation to determine the missing side length.

Practice

Source: Picture of Reliant Astrodome, SchuminWeb, Wikimedia Commons

1. The Astrodome in Houston has a diameter of 712 feet and a maximum height of 208 feet. Sabrina is constructing a scale model of the Astrodome using a scale factor of 40 feet to one inch. What will be the dimensions of Sabrina’s model?

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   Use the scale factor to reduce the dimensions of the actual Astrodome to the dimensions of the scale model.

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Source: Big Tex 2007 (1), Cracker989, Wikimedia Commons

2. Big Tex, at the Texas State Fairgrounds, is a statue that is 52 feet tall and has a sleeve length that is 22 feet long. Dashon built a replica that is 36 inches tall. What is the sleeve length of the replica that Dashon built?

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   Set up and solve an equation of proportions.

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   Height of Original over Height of Replica Height of Original Height of Replica = Height of Original over Height of Replica Sleeve Length of Original Sleeve Length of Replica
   
   52 ft over 3 ft 52 ft 3 ft = 22 ft over x ft 22 ft x ft
   
   x ≅ 1.27 ft ≅ 1 ft 3 1 over 4 1 24 in


3. The Port Isabel lighthouse near South Padre Island, Texas, is 57 feet tall and has a circumference around its base of about 60 feet. A scale model replica of the lighthouse is 18 inches tall. What scale factor was used to reduce the actual dimensions of the lighthouse to the dimensions of the replica?

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   Determine the number that could be multiplied by the original height in order to generate the height of the replica. Be sure that your units of measure are the same.

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   18 inches = 1.5 feet
   57 ft (x) = 1.5 ft
   
   x = 1.5 ft 57 ft
   x = 15 570 = 3 114