ACT® · Math: Geometry — Similarity, Right Triangles, and Trigonometry
Pythagorean Triples
A Pythagorean triple is a set of three whole numbers that satisfy a2 + b2 = c2, so right triangles built from them have no radicals, just clean integers.
Most right triangles give you ugly hypotenuses full of square roots. A handful don't, because their three sides happen to be whole numbers that satisfy a2 + b2 = c2 exactly. The most famous is 3-4-5: 32 + 42 = 9 + 16 = 25 = 52. These special triples show up constantly in geometry problems because they let the test give you a right triangle with tidy answers instead of forcing a radical.
The real power move is scaling. If 3-4-5 works, so does 6-8-10, 9-12-15, 30-40-50, or 3k-4k-5k for any whole number k, because multiplying every side by the same number multiplies a2, b2, and c2 by k2 too, so the equation still balances. The same goes for other base triples: 5-12-13, 8-15-17, 7-24-25, 20-21-29. Once you memorize the short list of base triples, you can recognize their scaled-up versions instantly instead of grinding through the Pythagorean theorem.
This matters most as a shortcut inside bigger problems: distance questions, diagonal-of-a-rectangle questions, or setups feeding into the two special right triangles (45-45-90 and 30-60-90) and trig ratios. Spotting a triple early saves time and avoids arithmetic mistakes with square roots, which is exactly why this skill sits right next to those other right-triangle tools.
Key terms
- Pythagorean triple
- A set of three positive whole numbers a, b, c satisfying a2 + b2 = c2.
- Scaling
- Multiplying every side of a triple by the same whole number k to produce a new, larger triple.
- Base triple
- The smallest version of a triple, like 3-4-5 or 5-12-13, from which all its scaled versions are built.
- Hypotenuse
- The longest side of a right triangle, opposite the right angle, always called c in a2 + b2 = c2.
Watch it explained
- How it worksHow does scaling 3-4-5 by any whole number generate an endless family of right triangles?
- Worked exampleFind the third side of a right triangle with legs 20 and 21 by spotting the triple.
- Worked exampleFind the hypotenuse of a right triangle with legs 24 and 45 without reaching for a calculator.
Check yourself
End of topic.