# 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.

**45**

^{0}-45^{0}-90^{0}TrianglesA 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**

Side 1: side 2 : hypotenuse = a : a: a√2

^{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.

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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

Special Right Triangles

30-60-90 Triangles

Pythagorean Theorem