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,

Write the formula for the area of a kite.

Plug in the given diagonals.

Pull out a common factor of two in and simplify.

Use the FOIL method to simplify.

Use the formula for perimeter to solve for the side length:

The Pythagorean Theorem states that . This question gives us the values of and , and asks us to solve for .

Take and and plug them into the equation as and :

Now we can start solving for :

The length of the hypotenuse is .

If creating a picture helps, draw a circle and place 8 at the top where 12 normally is. Then put 2, 4, and 6 at the positions of 3, 6, and 9 on a normal 12-hour analog clock. 1, 3, 5, and 7 go halfway in between each even number.

Now, because each section is equally spaced, and because there are 8 sections we simply divide the total number of degrees in a circle () by the number of sections (8). Thus:

Recall that the length of an arc is merely a percentage of the circumference. The circumference is found by the equation:

For our data, this is:

Now the percentage for our arc is based on the angle and the total degrees in a circle, namely, .

So, the length of the arc is:

Draw 3 radii, and then 3 new lines that bisect the radii. You get six 30-60-90 triangles.

These triangles can be used ot find a side length .

Using the formula for the area of an equilateral triangle in terms of its side, we get

To find the distance, first you need to find circumference. Thus,

Then, multiply the fraction of the clock they cover. Thus,

Based on the information given to you, you know that the area could be written as:

Likewise, you know that the perimeter is:

Now, isolate one of the values. For example, based on the second equation, you know:

Dividing everything by , you get:

Now, substitute this into the first equation:

To solve for , you need to isolate all of the variables on one side:

or:

Now, factor this:

, meaning that could be either or . These are the dimensions of your rectangle.

You could also get this answer by testing each of your options to see which one works for both the perimeter and the area.

Assigning the length of ED the value of x, the value of AE will be 3_x_. That makes the entire side AD equal to 4_x_. Since the figure is a square, all four sides will be equal to 4_x_. Also, since the figure is a square, then angle A of triangle ABE is a right angle. That gives triangle ABE sides of 3_x_, 4_x_ and 10. Using the Pythagorean theorem:

(3_x_)2 + (4_x_)2 = 102

9_x_2 + 16_x_2 = 100

25_x_2 = 100

_x_2 = 4

x = 2

With x = 2, each side of the square is 4_x_, or 8. The area of a square is length times width. In this case, that's 8 * 8, which is 64.