There are several accounts written about how the philosopher, scientist, and mathematician Thales succeeded in measuring the height of pyramids and other objects. By comparing the length of the shadow of an object with unknown height to the shadow of an object with a known height at the same time of day, Thales was able to use ratios to find the unknown height.

Click the play button below to explore the use of shadows to determine the height of a building. Use the animation to answer the questions that follow.

• As the sun moves, what happens to the lengths of the shadows?

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The lengths of the shadows increase.

• As the sun moves, what happens to the height of the building or the height of the student?

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Nothing. The height of the building and the height of the student remain constant.

• What is the ratio of the length of the student's shadow to the length of the building's shadow?

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1 over 12 1 12

• What shapes are created by the building and its shadow and the student and his shadow?

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Both pairs create similar right triangles.

• Set up and solve a proportion to determine b, the height of the building.

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### Pause and Reflect

For similar figures, the lengths of corresponding sides are proportional. Once you know the scale factor relating the lengths of corresponding sides, how can you use that scale factor to determine the length of a missing side?

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Because all pairs of corresponding sides have the same scale factor that relates their lengths, multiply or divide the length of the known side by the scale factor in order to determine the length of the corresponding side.

### Practice

1. Bernardo and his family visited the San Jacinto Monument near Deer Park, Texas. Bernardo is 5 feet 6 inches tall, and his sister measured his shadow to be 4 feet in length. At the same time, Bernardo and his sister measured the shadow of the monument to be 1371 over 2 1 2 yards. How tall is the San Jacinto Monument?

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Draw a figure showing the similar triangles, and label the known information. Set up and solve the proportional equation. Also be careful to make sure that your units are the same before any computation!

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2. Sheila visited her aunt in Paris, Texas. While she was there, she visited the replica of the Eiffel Tower. The tower cast a shadow that was 15 feet long, and Sheila cast a shadow that was 15 inches long. Sheila is 5 feet 8 inches tall. According to these measurements, how tall is the replica of the Eiffel Tower?

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Draw a figure showing the similar triangles, and label the known information. Set up and solve the proportional equation. Also be careful to make sure that your units are the same before any computation!

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Sources for images used in this section, as they appear, from top to bottom:

• Kheops Pyramid, Nina, Wikipedia
• Empire State Building From the Top of the Rock, Jiuguang Wang, Wikipedia
• Sequoiafarm Sequoiadendron giganteum, Svrrir Mirdsson, Wikipedia
• San Jacinto Monument, Matthiasb, Wikimedia Commons
• Anyjazz65 - Paris, Texas - Eiffel tower replica, MIRROR, Wikimedia Commons