In any right triangle, the square of the hypotenuse (the side across from the right angle, always the longest side) equals the sum of the squares of the other two sides, called the legs: a² + b² = c².
Finding the hypotenuse
Legs of 3 and 4: c² = 3² + 4² = 9 + 16 = 25, so c = √25 = 5.
Finding a missing leg
Hypotenuse 13, one leg 5. Rearrange to b² = c² − a²: b² = 169 − 25 = 144, so b = 12. Notice you subtract when the hypotenuse is the side you already know.
Is the triangle a right triangle?
The converse also holds: if a² + b² = c² with c the longest side, the triangle has a right angle. Sides 7, 24, 25: 49 + 576 = 625 = 25² ✓, so it is a right triangle. Sides 6, 7, 9: 36 + 49 = 85, but 9² = 81, so it is not.
Triples worth recognizing
- 3, 4, 5 (and multiples like 6, 8, 10)
- 5, 12, 13
- 8, 15, 17 (64 + 225 = 289 ✓)
- 7, 24, 25
The distance formula is the same idea
The distance between (x₁, y₁) and (x₂, y₂) is d = √((x₂ − x₁)² + (y₂ − y₁)²): the horizontal and vertical gaps are the legs. From (1, 2) to (4, 6): √(3² + 4²) = √25 = 5.
Common mistakes
- Using the theorem on a triangle with no right angle.
- Adding when the unknown is a leg (you should subtract).
- Forgetting the final square root and reporting c² instead of c.