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

## Example Questions

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### Example Question #1 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle

The perimeter of a 45-45-90 triangle is 100 inches. To the nearest tenth of an inch, what is the length of each leg?

Explanation:

Let  be the length of a leg; then the hypotenuse is , and the perimeter is

Therefore,

Explanation:

Explanation:

### Example Question #1 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle

Angle  in the triangle shown below is 45 degrees. Side  has a length of 10. What is the length of side ?

Explanation:

Since we know two of the three angles in this triangle, we can calculate the third, .

Therefore this is a 45/45/90 right triangle. Remember that 45/45/90 right triangles are have a leg:leg:hypotenuse ratio of 1:1:.

We know the hypotenuse, , so we can quickly calculate the length of one of the legs, , by dividing by :

To make this look like one of the answer choies, rationalize the denominator by muliplying the fraction by :

### Example Question #5 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle

is a  triangle.

What is the length of ?

There is not enough information given to answer this question.

Explanation:

We know that the sides of  triangles are in the ratio of , where the shorter sides lies opposite the  angles, and the longer side is the hypotenuse and lies opposite the right angle. We are given that the hypotenuse is .

Divide the length of the hypotenuse by  to calculate the ratio of magnification.

Multiply the length of the shorter sides by the ratio of magnification.

So the length of  (and ) is .

### Example Question #6 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle

The following image is not to scale.

Find the length of one of the legs of the right triangle.

Explanation:

Because of the tick marks on both legs, we can determine that this right triangle is a 45/45/90 triangle. Because the length of both legs are the same, this means that the angle opposite of each leg is also the same.

45/45/90 triangles are special, just like 30/60/90 triangles. Solving for one of the leg lengths can be determined easily through remembering the following:

Using this and the 7ft, we can solve for "s" which will provide us with the leg length.

while this is the correct answer, the options provided are represented as simplified radicals.

### Example Question #7 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle

If the hypotenuse of a right isosceles triangle is , what is the length of a side of the triangle?

Explanation:

A right isosceles triangle is also a  triangle.

To find the length of a side, we will need to use the Pythagorean Theorem:

Since this is an isosceles triangle,

The Pythagorean Theorem can then be rewritten as the following:

Since we are trying to find the length of a side of this triangle, solve for .

Simplify.

Multiply the fraction by one in the form of .

Solve.

Now, substitute in the length of the hypotenuse in for  to solve for the side of the triangle in the question.

Simplify.

Reduce.

### Example Question #8 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle

If the hypotenuse of a right isosceles triangle is , what is the length of a side of the triangle?

Explanation:

A right isosceles triangle is also a  triangle.

To find the length of a side, we will need to use the Pythagorean Theorem:

Since this is an isosceles triangle,

The Pythagorean Theorem can then be rewritten as the following:

Since we are trying to find the length of a side of this triangle, solve for .

Simplify.

Multiply the fraction by one in the form of .

Solve.

Now, substitute in the length of the hypotenuse in for  to solve for the side of the triangle in the question.

Simplify.

Reduce.

### Example Question #9 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle

If the hypotenuse of a right isosceles triangle is , what is the length of a side of this triangle?

Explanation:

A right isosceles triangle is also a  triangle.

To find the length of a side, we will need to use the Pythagorean Theorem:

Since this is an isosceles triangle,

The Pythagorean Theorem can then be rewritten as the following:

Since we are trying to find the length of a side of this triangle, solve for .

Simplify.

Multiply the fraction by one in the form of .

Solve.

Now, substitute in the length of the hypotenuse in for  to solve for the side of the triangle in the question.

Simplify.

Reduce.

### Example Question #2 : How To Find The Length Of The Side Of A 45/45/90 Right Isosceles Triangle

If the hypotenuse of a right isosceles triangle is , what is the length of a side of this triangle?

Explanation:

A right isosceles triangle is also a  triangle.

To find the length of a side, we will need to use the Pythagorean Theorem:

Since this is an isosceles triangle,

The Pythagorean Theorem can then be rewritten as the following:

Since we are trying to find the length of a side of this triangle, solve for .

Simplify.

Multiply the fraction by one in the form of .

Solve.

Now, substitute in the length of the hypotenuse in for  to solve for the side of the triangle in the question.

Simplify.

Reduce.

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