# ACT Math : Isosceles Triangles

## Example Questions

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

The area of an isosceles right triangle is .  What is its height that is correlative and perpendicular to a side that is not the hypotenuse?

Explanation:

Recall that an isosceles right triangle is a  triangle. That means that it looks like this:

This makes calculating the area very easy! Recall, the area of a triangle is defined as:

However, since  for our triangle, we know:

Now, we know that . Therefore, we can write:

Solving for , we get:

This is the length of the height of the triangle for the side that is not the hypotenuse.

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

What is the area of an isosceles right triangle that has an hypotenuse of length ?

Explanation:

Based on the information given, you know that your triangle looks as follows:

This is a  triangle. Recall your standard  triangle:

You can set up the following ratio between these two figures:

Now, the area of the triangle will merely be  (since both the base and the height are ). For your data, this is:

### Example Question #1 : 45/45/90 Right Isosceles Triangles

Find the height of an isoceles right triangle whose hypotenuse is

Explanation:

To solve simply realize the hypotenuse of one of these triangles is of the form where s is side length. Thus, our answer is .

### Example Question #1 : 45/45/90 Right Isosceles Triangles

The area of an isosceles right triangle is . What is its height that is correlative and perpendicular to this triangle's hypotenuse?

Explanation:

Recall that an isosceles right triangle is a  triangle. That means that it looks like this:

This makes calculating the area very easy! Recall, the area of a triangle is defined as:

However, since  for our triangle, we know:

Now, we know that . Therefore, we can write:

Solving for , we get:

However, be careful! Notice what the question asks: "What is its height that is correlative and perpendicular to this triangle's hypotenuse?" First, let's find the hypotenuse of the triangle. Recall your standard  triangle:

Okay, what you are actually looking for is  in the following figure:

Therefore, since you know the area, you can say:

Solving, you get:

### Example Question #1 : Isosceles Triangles

What is the area of an isosceles right triangle with a hypotenuse of ?

Explanation:

Now, this is really your standard  triangle. Since it is a right triangle, you know that you have at least one -degree angle. The other two angles must each be  degrees, because the triangle is isosceles.

Based on the description of your triangle, you can draw the following figure:

This is derived from your reference triangle for the  triangle:

For our triangle, we could call one of the legs . We know, then:

Thus, .

The area of your triangle is:

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

What is the area of an isosceles right triangle with a hypotenuse of ?

Explanation:

Now, this is really your standard  triangle. Since it is a right triangle, you know that you have at least one -degree angle. The other two angles must each be  degrees because the triangle is isosceles.

Based on the description of your triangle, you can draw the following figure:

This is derived from your reference triangle for the  triangle:

For our triangle, we could call one of the legs . We know, then:

Thus, .

The area of your triangle is:

### Example Question #3 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle

What is the area of an isosceles right triangle with a hypotenuse of ?

Explanation:

Now, this is really your standard  triangle. Since it is a right triangle, you know that you have at least one -degree angle. The other two angles must each be  degrees because the triangle is isosceles.

Based on the description of your triangle, you can draw the following figure:

This is derived from your reference triangle for the  triangle:

For our triangle, we could call one of the legs . We know, then:

Thus, .

The area of your triangle is:

### Example Question #4 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle

is a right isosceles triangle with hypotenuse . What is the area of ?

Explanation:

Right isosceles triangles (also called "45-45-90 right triangles") are special shapes. In a plane, they are exactly half of a square, and their sides can therefore be expressed as a ratio equal to the sides of a square and the square's diagonal:

, where  is the hypotenuse.

In this case,  maps to , so to find the length of a side (so we can use the triangle area formula), just divide the hypotenuse by :

So, each side of the triangle is  long. Now, just follow your formula for area of a triangle:

Thus, the triangle has an area of .

### Example Question #1 : 45/45/90 Right Isosceles Triangles

What is the perimeter of an isosceles right triangle with an hypotenuse of length ?

Explanation:

Your right triangle is a  triangle. It thus looks like this:

Now, you know that you also have a reference triangle for  triangles. This is:

This means that you can set up a ratio to find . It would be:

Your triangle thus could be drawn like this:

Now, notice that you can rationalize the denominator of :

Thus, the perimeter of your figure is:

### Example Question #2 : 45/45/90 Right Isosceles Triangles

What is the perimeter of an isosceles right triangle with an area of ?

Explanation:

Recall that an isosceles right triangle is also a  triangle. Your reference figure for such a shape is:

or

Now, you know that the area of a triangle is:

For this triangle, though, the base and height are the same. So it is:

Now, we have to be careful, given that our area contains . Let's use , for "side length":

Thus, . Now based on the reference figure above, you can easily see that your triangle is:
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