Question

which set of side lengths does not form a right triangle a.) 12, 16, 20 b.) 24, 32, 40 c.) 6, 8, 10 d.) 13, 16, 20 please thouroughly explain

Answer 1

The answer is d.) 13, 16, 20 since these do not satisfy the Pythagorean theorem. The other choices are all multiples of 3,4,5, which is a classic Pythagorean triangle.

Answer 2

A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are coprime. A right triangle whose sides form a Pythagorean triple is called a Pythagorean triangle.

The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a2 + b2 = c2; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the triangle with sides a = b = 1 and c = √2 is right, but (1, 1, √2) is not a Pythagorean triple because √2 is not an integer. Moreover, 1 and √2 do not have an integer common multiple because √2 is irrational.