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:

For your data, this is:

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 .

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:

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:

Therefore, your perimeter is:

This is a right triangle where the rope is the hypotenuse. One leg is the radius of the circle, 5 feet. The other leg is half of the tree's height, 12 feet. We can now use the Pythagorean Theorem  giving us . If  then .

The ratio of sides to hypotenuse of an isosceles right triangle is always . With this in mind, setting  as our hypotenuse means we must have leg lengths equal to:

Since the perimeter has two of these legs, we just need to multiply this by  and add the result to our hypothesis:

So, our perimeter in terms of  is: 

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.

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.

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:

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.