Recognizing special right triangles in geometry can help you to answer some questions quicker. A special right-triangle is a right triangle whose sides are in a particular ratio. You can also use the Pythagorean theorem, but if you can see that it is a special triangle it can save you some calculations.

There are two types of special triangles: 45^{0}-45^{0}-90^{0} triangles and 30^{0}-60^{0}-90^{0} triangles. 45^{0}-45^{0}-90^{0} Triangles

A 45^{0}- 45^{0}- 90^{0} triangle is a special right-triangle whose angles are 45^{0}, 45^{0}and 90^{0}. The lengths of the sides of a 45^{0}- 45^{0}- 90^{0} triangle are in the ratio of 1: 1:√2.

A right triangle with two sides of equal lengths is a 45^{0}- 45^{0}- 90^{0} triangle.
Side 1: side 2 : hypotenuse = a : a: a√2

Example 1:

Find x. Solution:
This is a right triangle with two equal sides so it must be a 45^{0}- 45^{0}- 90^{0} triangle.
So we use, a : a: a√2
Here a = 6
Hypotenuse = a√2 = 6√2.
_________________________________________________________________ Example 2 : Find the side of an isosceles right triangle if its hypotenuse is 10√2.

Solution:
Isosceles right triangle means 45^{0}- 45^{0}- 90^{0} triangle.
So we use, a : a: a√2
Here hypotenuse = a√2 = 10√2
Side = a = 10
_________________________________________________________________ Example 3 : Find the side of an isosceles right triangle if its hypotenuse is 18.

Solution:
Isosceles right triangle means 45^{0}- 45^{0}- 90^{0} triangle.
So we use, a : a: a√2
Here hypotenuse = a√2 = 18
Side = a = 18/√2
a= (18x√2)/(√2x√2) (Rationalize the denominator)
a = 18√2/2
a = 9√2
Each side = 9√2
Special Right Triangles