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45-45-90 Right Triangles

Leg times sqrt(2) equals hypotenuse.

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45-45-90 Right Triangles

45-45-90 Right Triangles

A right triangle with congruent legs and acute angles is an Isosceles Right Triangle. This triangle is also called a 45-45-90 triangle (named after the angle measures).

is a right triangle with , and .

45-45-90 Theorem: If a right triangle is isosceles, then its sides are in the ratio . For any isosceles right triangle, the legs are and the hypotenuse is always .

What if you were given an isosceles right triangle and the length of one of its sides? How could you figure out the lengths of its other sides?




Example 1

Find the length of .

Use the  ratio.

Here, we are given the hypotenuse. Solve for in the ratio.

Example 2

Find the length of , where  is the hypotenuse of a 45-45-90 triangle with leg lengths of .

Use the ratio.

Example 3

Find the length of the missing side.

Use the ratio. because it is equal to . So, .

Example 4

Find the length of the missing side.

Use the ratio. because it is equal to . So, .

Example 5

A square has a diagonal with length 10, what are the lengths of the sides?

Draw a picture.

We know half of a square is a 45-45-90 triangle, so .


  1. In an isosceles right triangle, if a leg is 4, then the hypotenuse is __________.
  2. In an isosceles right triangle, if a leg is , then the hypotenuse is __________.
  3. A square has sides of length 15. What is the length of the diagonal?
  4. A square’s diagonal is 22. What is the length of each side?

For questions 5-11, find the lengths of the missing sides. Simplify all radicals.

Review (Answers)

To see the Review answers, open this PDF file and look for section 8.5. 



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45-45-90 Theorem

For any isosceles right triangle, if the legs are x units long, the hypotenuse is always x\sqrt{2}.

45-45-90 Triangle

A 45-45-90 triangle is a special right triangle with angles of 45^\circ, 45^\circ, and 90^\circ.


The hypotenuse of a right triangle is the longest side of the right triangle. It is across from the right angle.

Legs of a Right Triangle

The legs of a right triangle are the two shorter sides of the right triangle. Legs are adjacent to the right angle.


The \sqrt{}, or square root, sign.

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