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Right Triangles and the Pythagorean Theorem

A free Geometry lesson from the “Triangles, Similarity, and Trigonometry” unit, with a worked example and practice problems including step-by-step solutions.

In a right triangle, the legs form the right angle and the hypotenuse is opposite the right angle. The Pythagorean theorem says a^2 + b^2 = c^2.

What you'll learn

Why it matters: Builders use the Pythagorean theorem to square corners, size ladders, and check diagonal braces.

Worked example

Problem. A right triangle has legs 6 and 8. Find the hypotenuse.

  1. Use a^2 + b^2 = c^2.
  2. 6^2 + 8^2 = 36 + 64 = 100.
  3. c = 10.

Answer: 10

Practice problems

1. A right triangle has legs 3 and 4. Find the hypotenuse.

Show solution
  1. Warm-up: First identify exactly what the question is asking: A right triangle has legs 3 and 4. Find the hypotenuse.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. 3^2 + 4^2 = 25.
  4. Square root of 25 is 5.
  5. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

2. A right triangle has legs 5 and 12. Find the hypotenuse.

Show solution
  1. Warm-up: First identify exactly what the question is asking: A right triangle has legs 5 and 12. Find the hypotenuse.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. 25 + 144 = 169.
  4. Square root of 169 is 13.
  5. Check the result by substituting or estimating: the response should match 13 and make sense in the original problem.

Answer: 13

3. A right triangle has hypotenuse 10 and one leg 6. Find the other leg.

Show solution
  1. Core Practice: First identify exactly what the question is asking: A right triangle has hypotenuse 10 and one leg 6. Find the other leg.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. 6^2 + b^2 = 10^2.
  4. b^2 = 64.
  5. Check the result by substituting or estimating: the response should match 8 and make sense in the original problem.

Answer: 8

4. A ladder reaches 12 feet high and its base is 5 feet from the wall. How long is the ladder?

Show solution
  1. Core Practice: First identify exactly what the question is asking: A ladder reaches 12 feet high and its base is 5 feet from the wall. How long is the ladder?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. The ladder is the hypotenuse.
  4. 5^2 + 12^2 = 169.
  5. Check the result by substituting or estimating: the response should match 13 and make sense in the original problem.

Answer: 13

5. Which side lengths form a right triangle?

Choices: 7, 24, 25 · 5, 5, 12 · 6, 8, 12 · 4, 9, 10

Show solution
  1. Challenge: First identify exactly what the question is asking: Which side lengths form a right triangle?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. 7^2 + 24^2 = 25^2.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 7, 24, 25

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