The Pythagorean Theorem
A free Pre-Algebra lesson from the “Geometry Foundations” unit, with a worked example and practice problems including step-by-step solutions.
In a right triangle, the side opposite the right angle is the hypotenuse (c). The other two sides are legs (a and b). The Pythagorean Theorem says a^2 + b^2 = c^2. To find a missing side, substitute the known sides and solve. In Geometry Foundations, the goal is not just to get an answer but to recognize the structure of the problem, choose a reliable strategy, and explain why the result is reasonable. The practice set now includes targeted skill work, transfer questions, and mixed review so students build fluency and retention.
What you'll learn
- Use a^2 + b^2 = c^2 to find a missing hypotenuse
- Use the theorem in reverse to find a missing leg
- Recognize common Pythagorean triples
Worked example
Problem. A right triangle has legs 5 and 12. Find the hypotenuse.
- Use a^2 + b^2 = c^2 with a=5, b=12.
- 25 + 144 = c^2, so c^2 = 169.
- c = sqrt(169) = 13.
- Connect the calculation back to The Pythagorean Theorem so the method, not just the arithmetic, is clear.
Answer: 13
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. Right triangle with legs 3 and 4. Find the hypotenuse.
Show solution
- 9 + 16 = 25.
- sqrt(25) = 5.
Answer: 5
2. Right triangle with legs 6 and 8. Find the hypotenuse.
Show solution
- 36 + 64 = 100.
- sqrt(100) = 10.
Answer: 10
3. Right triangle with legs 9 and 12. Find the hypotenuse.
Show solution
- 81 + 144 = 225.
- sqrt(225) = 15.
Answer: 15
4. Hypotenuse 13, one leg 5. Find the other leg.
Show solution
- Other leg squared = 169 - 25 = 144.
- sqrt(144) = 12.
Answer: 12
5. Hypotenuse 17, one leg 8. Find the other leg.
Show solution
- Other leg squared = 289 - 64 = 225.
- sqrt(225) = 15.
Answer: 15
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