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Pythagorean Theorem and its Converse

Overview

The Pythagorean Theorem relates the three sides of a right triangle, and its converse turns the relationship around: given three lengths, you can test whether they form a right triangle at all. This lesson finds missing sides of right triangles — exactly, in simplest radical form, or rounded — and then uses the converse, including a comparison of squares, to classify triangles as right, acute, or obtuse.

Vocabulary

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.
Legs
The two sides of a right triangle that form the right angle.
Hypotenuse
The side of a right triangle opposite the right angle. It is always the longest side, and it is the side labeled c in a² + b² = c².
Converse of the Pythagorean Theorem
If three lengths satisfy a² + b² = c², with c the largest length, then the lengths form a right triangle.
Simplest Radical Form
A radical expression simplified so that no perfect-square factor remains under the radical sign, such as writing √50 as 5√2.

Main Ideas

  1. The theorem only works for right triangles, and the largest value always goes in for c, the hypotenuse.
  2. To find a missing hypotenuse, add the squares of the legs and take the square root. To find a missing leg, subtract the squares and take the square root.
  3. Answers may be left in simplest radical form or rounded as directed — a rounded answer should say so (for example, "to the nearest tenth").
  4. Classification test with c the largest side: if a² + b² = c² the triangle is right; if a² + b² > c² it is acute; if a² + b² < c² it is obtuse — a c² that is "too big" pushes the angle opposite it open.
  5. Familiar triples such as 3-4-5, 5-12-13, and 8-15-17 (and their multiples) satisfy the theorem exactly and are worth recognizing on sight.

Lesson Notes

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Notes page 1

Lesson Video

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Practice Problems

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Download the practice worksheet for this lesson, work the problems, then check your work with the solutions.