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.

The answer can be found two different ways. The first step is to realize that this is really a triangle question, even though it starts with a square. By drawing the square out and adding the diagonal, you can see that you form two right triangles. Furthermore, the diagonal bisects two ninety-degree angles, thereby making the resulting triangles a  triangle. 

From here you can go one of two ways: using the Pythagorean Theorem to find the diagonal, or recognizing the triangle as a  triangle.

1) Using the Pythagorean Theorem

Once you recognize the right triangle in this question, you can begin to use the Pythagorean Theorem. Remember the formula: , where  and  are the lengths of the legs of the triangle, and  is the length of the triangle's hypotenuse.

In this case, . We can substitute these values into the equation and then solve for , the hypotenuse of the triangle and the diagonal of the square:

The length of the diagonal is .

2) Using Properties of  Triangles

The second approach relies on recognizing a  triangle. Although one could solve this rather easily with Pythagorean Theorem, the following method could be faster.

 triangles have side length ratios of , where  represents the side lengths of the triangle's legs and  represents the length of the hypotenuse.

In this case,  because it is the side length of our square and the triangles formed by the square's diagonal.

Therefore, using the  triangle ratios, we have  for the hypotenuse of our triangle, which is also the diagonal of our square.

Recall that an isosceles right triangle is also a  triangle. It has sides that appear as follows:

Therefore, the area of the triangle is:

, since the base and the height are the same.

For our data, this means:

Solving for , you get:

So, your triangle looks like this:

Now, you can solve this with a ratio and easily find that it is .  You also can use the Pythagorean Theorem. To do the latter, it is:

Now, just do your math carefully:

That is a weird kind of factoring, but it makes sense if you distribute back into the group. This means you can simplify:

The pole and its shadow make a right angle. Because they are the same length, they form an isosceles right triangle (45/45/90). We can use the Pythagorean Theorem to find the hypotenuse. . In this case, . Therefore, we do . So . 

To solve, simply use the Pythagorean Theorem.

Recall that an isosceles right triangle has two leg lengths that are equal.

Therefore, to solve for the hypotenuse let  and  in the Pythagorean Theorem.

Thus,