Drawing the diagonal of a square decomposes the square into two congruent triangles. They are right isosceles triangles with acute angles of 45 degrees. These congruent angles make all right isosceles triangles similar by the Angle-Angle Triangle Similarity Theorem.

Consider an isosceles right triangle with legs 1 unit long, where is the length of the hypotenuse. By the Pythagorean Theorem, we can say , so . The hypotenuse of an isosceles right triangle with legs 1 unit long is  units long.

Now, consider an isosceles right triangle with legs  units long. By the Angle-Angle Triangle Similarity Theorem, the triangle is similar to the isosceles right triangle with side lengths of 1, 1, and  units. A scale factor of  takes the triangle with a leg length of 1 unit to the triangle with a leg length of . Therefore, the hypotenuse of the isosceles right triangle with legs  units long is  units long. 

Triangle. An angle is marked as a right angle. The other two angles are labeled as 45 degrees each. The sides opposite the 45 degree angles are labeled x. The side opposite the right angle is labeled x times square root of 2.

In triangle , so  is 6 units long and is  units long. 

In triangle , so , which means both and are  units long.