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Keywords: Right triangle,isosceles right triangle,special right triangle
Common Core Standard  G.SRT.6
A triangle with two equal sides, and a ninety degree angle will be a 45 45 90 triangle. Notice the triangle drawn inside a circle is a 45 45 90 because the radii are equal, and there is a 90 degree angle.
45-45-90 Triangle
45 45 90
45 45 90
 As the name implies a  45-45-90 triangle has two angle measures of 45 degrees, and one of ninety degrees.

A forty five,forty five, ninety triangle has two equal sides.

The hypotenuse is always the longest side, and is opposite the right angle.

The length of the hypotenuse equals, leg length√2

The length of one leg of a 45-45-90 triangle equals, hypotenuse/√2

A helpful angle to side ratio to remember when working with 45 45 90 triangles is as follows:

45  45  90
x     x    x√2

The legs opposite the forty five degree angles are equal, and the hypotenuse which is opposite the 90 degree angle, is equal to x√2  

You can also write this ratio as 1:1: √2

The formula for finding area equals ½(leg)2

A 45 45 90 triangle is also called an isosceles right triangle

The diagonal of a square creates two 45 45 90 triangles.

Questions answered in this video

If given the hypotenuse how do you find the leg length of a 45 45 90 triangle?

If given the leg length how do you find the hypotenuse of a 45 45 90 triangle?

What shortcuts can I use to find the leg lengths of a special right triangle easily?

Example 2. Find the length of the two sides of a 45-45-90 triangle with a hypotenuse length of 12 units.

Example 1: How to solve a 45 45 90 triangle. What is the length of the hypotenuse of a 45-45-90 triangle with a leg length of 7 units?

Step 1. Use x√(2 )    x = leg length 

Step 2. 7√(2 ) units

right trianle 7x7/special right triangles 45 45 90

Identifying 45 45 90 triangles

equation 12 divided by square root 2
Hypotenuse =
90 degree triangle with a hypotenuse of 12 units
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Properties of a special right triangle 45 45 90

properties 45 45 90 triangle

Shortcuts for finding 45 45 90 Triangle sides

We all love shortcuts. Try these for finding the side lengths of a 45 45 90 special right triangle.

Just remember √2

Hypotenuse = leg length x √2

Leg length = hypotenuse/√2
45 45 90 Triangles are special right triangles that have unique properties that allow one to find missing angles and side lengths.