The most famous result in all of geometry applies specifically to right triangles. The Pythagorean Theorem states that in any right triangle with legs of length a and b and hypotenuse of length c, the equation a² + b² = c² holds true. The hypotenuse is the side opposite the right angle, and it is always the longest side of the triangle. A classic example uses legs of length 3 and 4: 3² + 4² = 9 + 16 = 25, so c = √25 = 5. The triple (3, 4, 5) is the most well-known Pythagorean triple, but others include (5, 12, 13) and (8, 15, 17).
The Pythagorean Theorem also has a useful converse. If the square of the longest side of a triangle equals the sum of the squares of the other two sides, then the triangle is a right triangle. Even when the relationship does not hold exactly, we can use it to classify a triangle. If c² < a² + b², where c is the longest side, the triangle is acute. If c² > a² + b², the triangle is obtuse. This makes the Pythagorean relationship a powerful diagnostic tool for any triangle.
Two special right triangles appear so often that their side ratios are worth memorizing. In a 45-45-90 triangle, the two legs are equal and the sides are in the ratio 1 : 1 : √2. So if each leg has length x, the hypotenuse is x√2. In a 30-60-90 triangle, the sides are in the ratio 1 : √3 : 2. The side opposite 30° is the shortest, the side opposite 60° is √3 times that length, and the hypotenuse is twice the shortest side. Knowing these shortcuts saves considerable time when solving problems involving right triangles.