Pythagorean Theorem
Overview
The Pythagorean Theorem applies to right triangles only: the squares of the two legs add to the square of the hypotenuse. It finds a missing side when the other two are known — a leg is found by subtracting, the hypotenuse by adding. Whole-number side sets that satisfy the theorem are Pythagorean triples, and every triple belongs to a family generated by a root triple.
Vocabulary
- Right Triangle
- A triangle that contains one right angle (an angle measuring 90°).
- Legs
- The two sides of a right triangle that form the right angle.
- Hypotenuse
- The side of a right triangle opposite the right angle. The hypotenuse is always the longest side of the triangle.
- Pythagorean Theorem
- In any right triangle, the sum of the squares of the measures of the legs is equal to the square of the measure of the hypotenuse: a² + b² = c², where a and b are legs and c is the hypotenuse.
- Pythagorean Triple
- Three whole numbers that satisfy the Pythagorean Theorem, such as 3, 4, 5 or 5, 12, 13.
- Root Triple
- A Pythagorean triple whose numbers have a greatest common factor of 1. Every multiple of a root triple is another triple in the same family.
Main Ideas
- The theorem is only for right triangles, and c always names the hypotenuse — identify it before substituting.
- Finding the hypotenuse: add the squares of the legs, then take the square root. Finding a leg: subtract the square of the known leg from the square of the hypotenuse, then take the square root.
- Answers may be left in simplest radical form; a simplified radical keeps the exact value.
- Recognizing common triples (3-4-5, 5-12-13, 7-24-25, 8-15-17) and their multiples speeds up every problem.
- Always check that the hypotenuse comes out longest, and that lengths are positive.
Lesson Notes
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Lesson Video
Watch the lesson video.
Interactive Activity
Explore the theorem with squares on the sides, build missing sides, and sort out Pythagorean triples.
Practice Problems
Want More Practice?
Download the practice worksheet for this lesson, work the problems, then check your work with the solutions.