First draw the triangle with one leg as the base. Since this is an isosceles triangle both legs measure 12cm. Label each leg with its measure of 12cm. One of the legs will act as the base, and the other leg will act as the height. In this problem we don't need to worry about the hypotenuse.

We will need to use our formula for area of a triangle 

When we apply our formula we have half of twelve multiplied by twelve

since half of twelve is six, the next step is to multiply 12 times 6, and we get 72.

Since this triangle is measured in centimeters and we multiplied to get the area, our answer will be in centimeters squared. 

Notice that the given triangle is a right isosceles triangle. The hypotenuse of the triangle is the same as the diameter of the circle; therefore, we can use the Pythagorean theorem to find the length of the legs of this triangle.

Substitute in the given hypotenuse to find the length of the leg of a triangle.

Simplify.

Now, recall how to find the area of a triangle.

Since we have a right isosceles triangle, the base and the height are the same length.

Solve.

Isosceles right triangles are special triangles because they possess angles of the following measures: , , and . These triangles are known as 45-45-90 triangles and have special characteristics. Recall the Pythagorean theorem:

In this equation,  is the length of the triangle's base,  is equal to its height, and  is equal to the length of its hypotenuse. In an isosceles right triangle, the base and the height have the same length; therefore,  is equal to , and you can rewrite the Pythagorean theorem like this:

Rearrange the equation so that  is isolated on one side of the equals sign. First, simplify by dividing both sides of the equation by 2.

Next, take the square root of both sides.

Now, plug in the value of the hypotenuse to find the height/base of the given triangle.

Now, recall how to find the area of a triangle:

Since the base and the height are the same length, we can find the area of the given triangle.

Solve.

An isoceles right triangle has two congruent legs.  We use the Pythagorean Theorem, which states that a² + b² = c², where a and b are the legs and c is the hypotenuse.

Let = leg length.

Because this is an isosceles triangle, the two legs have the same length. Plug this and the hypotenuse length into the Pythagorean Theorem and solve for x:

Thus the perimeter is .

Plug in our value for x:

The perimeter of a triangle is the distance around the outside, and can be found by adding the side lengths together: .

The first point of business is to realize is that we have a 45-45-90 triangle. That means the two legs of our triangle are congruent and are thus each 8. We can find our hypotenuse by multiplying by . Thus our hypotenuse is .

Our perimeter is simply the sum of of the lengths of our three sides.

We are given that  is a  triangle and that . Calculate the length of sides  and  using either the Pythagorean Theorem or the  ratio of the sides of a  triangle.

Using the Pythagorean Theorem:

Because  is a  triangle, we know  so we can represent both legs of the triangle with one variable.  Let's use .

Using the ratio of the sides:

Divide the length of the hypotenuse by .

Either way, the length of 

Calculate the perimeter by adding the length of all the sides.

This can be factored to .

The problem tells us the triangle is 45/45/90. The goal is to solve for the perimeter, which can be determined through , where the s's are in reference to the three sides and P stands for perimeter. 

In the figure, two of the three sides are given. In order to calculate the hypotenuse, two methods are possible:

1. using the Pythagorean Theorem 

2. using 

After calculations, the hypotenuse is 

Perimeter can be calculated out to be:

In order to solve for the perimter (the sum of all sides), all side lengths must be known. 

Because it's been stated the triangle is 45/45/90, this means that it is also isosceles. Therefore, given that one of the leg lengths is 5 cm, this means that the other leg must also be 5 cm. This leaves the hypotenuse as unknown; let's label this as x. 

The third side can be easily determined through the Pythagorean Theorem because it's a right triangle. 

, c=x

But because the hypotenuse measures distance, x cannot be a negative number. Therefore, x=5√2.

Now, perimeter can be solved for.

To find the perimeter we must find the hypotenuse and then sum all side lengths to find the perimeter. Remember the hypostenuse of an isoceles right triangle is the side length multiplied by the square root of .

Thus,