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45-45-90 Triangles

Wow. What a square.

Much better.

Now we have two 45-45-90 isosceles triangles, with diagonal d as our hypotenuse. A little known fact about these triangles: they're more magical than Gandalf from The Lord of the Rings. Don't believe us?

See? Wizardly. They have special properties, too. If we throw the Pythagorean Theorem at them, we'll see why.

a2 + b2 = c2

Since c is d, we can substitute that in. The magical thing about these triangles, though, is that a and b equal each other. In our case, they're both x.

d 2 = x2 + x2
d 2 = 2x2

Taking the positive square root of both sides will tell us what the hypotenuse equals.

Factoring whatever's under the square root will simplify things. Watch carefully. Things are about to get magical.

One side to find them all:
One leg times root 2
Equals the hypotenuse
And that, we know, is true.

For every 45-45-90 triangle, one side can give us the lengths of all the sides. That's magical enough to make J.R.R. Tolkien proud.

Sample Problem

What is the length of this triangle's legs?

Its hypotenuse is , and like the magic poem says, we only need one side to figure out the others. As usual, we'll start with the formula we derived from the Pythagorean Theorem.

Substitute in our hypotenuse for d.

Divide both sides by .

x = 2

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