How to find the perimeter of a 45/45/90 right isosceles triangle

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: