Pythagorean Theorem

45-45-90 Triangles

A 45-45-90 triangle is an isosceles right triangle whose angles are 45º, 45º and 90º.

In a 45-45-90 triangle you can use Pythagorean Theorem to find any missing leg of the triangle:

A leg of a 45-45-90 triangle measures 5 units. Use the Pythagorean Theorem to show that the hypotenuse of the triangle measures units.

Solution:
The Pythagorean Theorem states that for a right triangle, a²+b²=c², where a and b are the lengths of the legs and c is the length of the hypotenuse. Because the triangle is a 45-45-90 triangle, its legs are congruent, and the value of both a and b is 5 units. Use this information to solve for c, the length of the hypotenuse.

a²+b²=c²

5²+5²=c²	Substitute
50=c²	Simplify
	Take the square root of each side
	Simplify

The length of the hypotenuse is equal to the length of a leg multiplied by , or units.

But a special relationship exists among the lengths of the sides of a 45-45-90 triangle. Knowing the relationship between the side lengths of these triangles, may save your time on any test.

Relationship between the side lengths of 45-45-90 triangles

Using c for the hypotenuse, each leg is
The hypotenuse is times bigger than the sides:

(When the legs of a 45-45-90 triangle are both 1, the hypotenuse is . So more generally, the two legs and hypotenuse of a 45-45-90 triangle are in a ratio of s: )

Find the length of the hypotenuse of a 45-45-90 triangle if the length of the other two sides are both 5 inches.

Solution:
You are given that the both sides are 5. Using the relationship between the side lengths of 45-45-90 triangles, the length of the hypotenuse is inches.