The Pythagorean theorem and its applications

In a right triangle the square of the hypotenuse equals the sum of the squares of the legs: c² = a² + b². The converse: if in a triangle the square of the longest side equals the sum of the squares of the other two, the triangle is right-angled. Integer triples such as 3, 4, 5 and 5, 12, 13 are called Pythagorean triples. The leg opposite a 30° angle is half the hypotenuse, and the height of an equilateral triangle with side a is a√3/2. When solving, we take the square root and keep only the positive value because a length is positive.