The Pythagorean Theorem

The Theorem

The Pythagorean Theorem applies to any right triangle — a triangle with one 90^{\circ} angle. Label the two shorter sides (the legs) a and b, and the longest side (the hypotenuse, opposite the right angle) c. Then:

Picture a right triangle with the right angle at the bottom-left corner, a horizontal leg of length a along the bottom, a vertical leg of length b along the left side, and the hypotenuse c slanting from the top of the vertical leg to the end of the horizontal leg.

Common Pythagorean Triples

A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy a^2+b^2=c^2. Recognizing these instantly saves time on contest problems:

• 3-4-5: 3^2+4^2 = 9+16=25=5^2
• 5-12-13: 5^2+12^2=25+144=169=13^2
• 8-15-17: 8^2+15^2=64+225=289=17^2

Any whole-number multiple of a triple is also a triple — for instance, doubling 3-4-5 gives 6-8-10, which also satisfies the theorem.

The Converse: Testing for a Right Triangle

The converse of the Pythagorean Theorem is just as useful: if the three side lengths of a triangle satisfy a^2+b^2=c^2 (where c is the longest side), then the triangle must be a right triangle. This gives a quick test — no protractor needed, just arithmetic.

You can also use this comparison to determine if an unknown triangle is acute or obtuse: if a^2+b^2 > c^2, the triangle is acute; if a^2+b^2 < c^2, the triangle is obtuse (where c is always the longest side).

Example

A right triangle has one leg of length 9 and a hypotenuse of length 15. Find the length of the other leg.

1. Set up the theorem with a=9 and c=15: 9^2 + b^2 = 15^2
2. Simplify: 81 + b^2 = 225
3. Solve: b^2 = 144 \implies b = 12
4. Notice this is just the 3-4-5 triple scaled by 3: (9,12,15) = 3\times(3,4,5), confirming the answer.

Key Takeaways

• a^2+b^2=c^2, where c is the hypotenuse of a right triangle.
• Memorize 3-4-5, 5-12-13, and 8-15-17 (and their multiples) to skip arithmetic on contests.
• The converse lets you confirm a triangle is right just from its three side lengths.
• Comparing a^2+b^2 to c^2 tells you if a triangle is acute, right, or obtuse.