454590 Triangles
A 454590 triangle is an isosceles right triangle whose angles are 45º, 45º and 90º.
In a 454590 triangle you can use Pythagorean Theorem to find any missing leg of the triangle:
A leg of a 454590 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, a^{2}+b^{2}=c^{2}, where a and b are the lengths of the legs and c is the length of the hypotenuse. Because the triangle is a 454590 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.
a^{2}+b^{2}=c^{2}
5^{2}+5^{2}=c^{2}

Substitute

50=c^{2}

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 454590 triangle. Knowing the relationship between the side lengths of these triangles, may save your time on any test.
Find the length of the hypotenuse of a 454590 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 454590 triangles, the length of the hypotenuse is inches.