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How to Work with 45-45-90-Degree Triangles

All 45-45-90-degree triangles (also known as 45ers) have sides that are in a unique ratio. The two legs are the exact same length, and the hypotenuse is that length times the square root of 2. The figure shows the ratio. (If you look at the 45er triangle in radians, you have

image0.png

Either way, it's still the same ratio.)

A 45-45-90-degree right triangle.
A 45-45-90-degree right triangle.

Why is this triangle important? Because any time you're given one side of a 45er triangle, you can figure out the other two sides. When you are asked to complete calculations with this type of triangle, it will fall into one of two categories:

  • Type 1: You're given one leg.

    Because you know both legs are equal, you know the length of both the legs. You can find the hypotenuse by multiplying this length by the square root of 2.

  • Type 2: You're given the hypotenuse.

    Divide the hypotenuse by the square root of 2 to find the legs (which are equal).

Here's an example calculation: The diagonal in a square is 16 centimeters long. How long is each side of the square? Draw it out first. The figure shows the square.

A square with a diagonal.
A square with a diagonal.

The diagonal of a square divides the angles into 45-degree pieces, so you have the hypotenuse of a 45er triangle. To find the legs, divide the hypotenuse by the square root of 2. When you do, you get

image3.png

You must now rationalize the denominator to get

image4.png

which is the measure of each side of the square.

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