# Special Right Triangles

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.
450-450-900 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 450- 450- 900 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.
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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
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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

Special Right Triangles
30-60-90 Triangles
Pythagorean Theorem