Pythagorean Triples

Pythagorean Triples

How to use the Pythagorean triples to solve the sides of a given right triangle: commonly used triples, examples, and their solutions.

Triples

The Pythagorean triples are three positive integers that satisfy the Pythagorean theorem. (3, 4, 5), (5, 12, 13), (7, 24, 25)

The Pythagorean triples are three positive integers
that satisfy the Pythagorean theorem:
a2 + b2 = c2.

Below triples are the commonly used triples.

(3, 4, 5): 32 + 42 = 52
(5, 12, 13): 52 + 122 = 132
(7, 24, 25): 72 + 242 = 252

Example 1

Find the value of x.

Starting from the shortest side,
the sides of the given right triangle are (x, 4, 5).

Then the related triple is (3, 4, 5).

So x = 3.

Example 2

Find the value of x.

Starting from the shortest side,
the sides of the given right triangle are (5, 12, x).

Then the related triple is (5, 12, 13).

So x = 13.

Example 3

Find the value of x.

Starting from the shortest side,
the sides of the given right triangle are (6, x, 10).

Then the related triple is (3, 4, 5).

So draw a (3, 4, 5) right triangle
next to the given triangle.

These two triangles are similar.

Since these two triangles are similar,
their sides are proportional.

So x/4 = 6/3.

Similar triangles

6/3 = 2

Multiply 4 on both sides.

Then x = 8.