Applications of the Pythagorean Theorem

Examples, videos, and solutions to help Grade 8 students learn how to use the Pythagorean theorem to determine missing side lengths of right triangles.

New York State Common Core Math Module 2, Grade 8, Lesson 16

Worksheets for Grade 8

Lesson 16 Student Outcomes

• Students use the Pythagorean theorem to determine missing side lengths of right triangles.

Lesson 16 Summary
The Pythagorean theorem can be used to find the unknown length of a leg of a right triangle.
An application of the Pythagorean theorem allows you to calculate the length of a diagonal of a rectangle, the distance between two points on the coordinate plane and the height that a ladder can reach as it leans against a wall.

Classwork
Exercises and Examples
Example 1
Pythagorean theorem as it applies to missing side lengths of triangles:
Given a right triangle with a hypotenuse with length units and a leg with length units, as shown, determine the length of the other leg.

Example 2
Pythagorean theorem as it applies to missing side lengths of triangles in a real-world problem:
• Suppose you have a ladder of length 13 feet. To make it sturdy enough to climb you must place the ladder exactly 5 feet from the wall of a building. You need to post a banner on the building 10 feet above the ground. Is the ladder long enough for you to reach the location you need to post the banner?

Example 3
Pythagorean theorem as it applies to missing side lengths of a right triangle:
• Given a right triangle with a hypotenuse of length 15 units and a leg of length 9, what is the length of the other leg?

Exercises

1. Use the Pythagorean theorem to find the missing length of the leg in the right triangle.
2. You have a 15-foot ladder and need to reach exactly 9 feet up the wall. How far away from the wall should you place the ladder so that you can reach your desired location?

Example 4
Pythagorean theorem as it applies to distances on a coordinate plane:
• We want to find the length of the segment AB on the coordinate plane, as shown.

Example 5
Pythagorean theorem as it applies to the length of a diagonal in a rectangle:
• Given a rectangle with side lengths of 8 cm and 2 cm, as shown, what is the length of the diagonal?

Exercises 3 - 6
3. Find the length of the segment AB. 4. Given a rectangle with dimensions 5 cm and 10 cm, as shown, find the length of the diagonal. 5. A right triangle has a hypotenuse of length 13 in and a leg with length 4 in. What is the length of the other leg? 6. Find the length of in the right triangle below.

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