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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. In Triangles, Similarity, and Trigonometry, students need to read the diagram, name the relationship, choose a theorem or formula, and justify why the result follows. The expanded practice now includes fluency, transfer, cumulative review, and proof-style reasoning so Geometry feels connected instead of isolated by topic.

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.
  4. Connect the result back to Right Triangles and the Pythagorean Theorem so the geometric relationship is explicit.

Answer: 10

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

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

Show solution
  1. 3^2 + 4^2 = 25.
  2. Square root of 25 is 5.

Answer: 5

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

Show solution
  1. 25 + 144 = 169.
  2. Square root of 169 is 13.

Answer: 13

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

Show solution
  1. 6^2 + b^2 = 10^2.
  2. b^2 = 64.

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. The ladder is the hypotenuse.
  2. 5^2 + 12^2 = 169.

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. 7^2 + 24^2 = 25^2.

Answer: 7, 24, 25

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