# ACT Math : How to find the length of the side of a 45/45/90 right isosceles triangle

## Example Questions

### Example Question #13 : Isosceles Triangles

A 44/45/90 triangle has a hypotenuse of . Find the length of one of its legs.    Cannot be determined Explanation:

It's helpful to remember upon coming across a 45/45/90 triangle that it's a special right triangle. This means that its sides can easily be calculated by using a derived side ratio: Here, represents the length of one of the legs of the 45/45/90 triangle, and represents the length of the triangle's hypotenuse. Two sides are denoted as congruent lengths ( ) because this special triangle is actually an isosceles triangle. This goes back to the fact that two of its angles are congruent.

Therefore, using the side rules mentioned above, if , this problem can be resolved by solving for the value of :   Therefore, the length of one of the legs is 1.

### Example Question #14 : Isosceles Triangles

In a 45-45-90 triangle, if the hypothenuse is long, what is a possible side length?      Explanation:

If the hypotenuse of a 45-45-90 triangle is provided, its side length can only be one length, since the sides of all 45-45-90 triangles exist in a defined ratio of , where represents the length of one of the triangle's legs and represents the length of the triangle's hypotenuse. Using this method, you can set up a proportion and solve for the length of one of the triangle's sides: Cross-multiply and solve for .  Rationalize the denominator. You can also solve this problem using the Pythagorean Theorem. In a 45-45-90 triangle, the side legs will be equal, so . Substitute for and rewrite the formula. Substitute the provided length of the hypothenuse and solve for .    While the answer looks a little different from the result of our first method of solving this problem, the two represent the same value, just written in different ways. ### Example Question #201 : Act Math

In a triangle, if the length of the hypotenuse is , what is the perimeter?     Explanation:

1. Remember that this is a special right triangle where the ratio of the sides is: In this case that makes it: 2. Find the perimeter by adding the side lengths together: ### Example Question #16 : Isosceles Triangles

The height of a triangle is . What is the length of the hypotenuse?     Explanation:

Remember that this is a special right triangle where the ratio of the sides is: In this case that makes it: Where is the length of the hypotenuse.

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