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
Practice problems
1. Right triangle with legs 3 and 4. Find the hypotenuse.
Show solution
- Warm-up: First identify exactly what the question is asking: Right triangle with legs 3 and 4. Find the hypotenuse.
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- 9 + 16 = 25.
- sqrt(25) = 5.
- Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.
Answer: 5
2. Right triangle with legs 6 and 8. Find the hypotenuse.
Show solution
- Warm-up: First identify exactly what the question is asking: Right triangle with legs 6 and 8. Find the hypotenuse.
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- 36 + 64 = 100.
- sqrt(100) = 10.
- Check the result by substituting or estimating: the response should match 10 and make sense in the original problem.
Answer: 10
3. Right triangle with legs 9 and 12. Find the hypotenuse.
Show solution
- Warm-up: First identify exactly what the question is asking: Right triangle with legs 9 and 12. Find the hypotenuse.
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- 81 + 144 = 225.
- sqrt(225) = 15.
- Check the result by substituting or estimating: the response should match 15 and make sense in the original problem.
Answer: 15
4. Hypotenuse 13, one leg 5. Find the other leg.
Show solution
- Core Practice: First identify exactly what the question is asking: Hypotenuse 13, one leg 5. Find the other leg.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Other leg squared = 169 - 25 = 144.
- sqrt(144) = 12.
- Check the result by substituting or estimating: the response should match 12 and make sense in the original problem.
Answer: 12
5. Hypotenuse 17, one leg 8. Find the other leg.
Show solution
- Core Practice: First identify exactly what the question is asking: Hypotenuse 17, one leg 8. Find the other leg.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Other leg squared = 289 - 64 = 225.
- sqrt(225) = 15.
- Check the result by substituting or estimating: the response should match 15 and make sense in the original problem.
Answer: 15
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