The inscribed rectangle is a 20 by 20 square. Since opposite sides of the square are parallel, the corresponding angles of the two smaller right triangles are congruent; therefore, the two triangles are similar and, by definition, their sides are in proportion.

The small top triangle has legs 10 and 20. Therefore, the length of its hypotenuse can be determined using the Pythagorean Theorem:

The small top triangle has short leg 10 and hypotenuse . The blue triangle has short leg 20 and unknown hypotenuse , where  can be calculated with the proportion statement

By the Law of Sines,

Substitute  and solve for :

By the Law of Cosines,

Substitute :

Two sides of the triangle formed measure 4 each; the included angle is one angle of the regular pentagon, which measures

The length of the third side can be found by applying the Law of Cosines:

where :

In a triangle, the shortest side is opposite the angle of least measure; the longest side is opposite the angle of greatest measure. Therefore, if we order the angles, we can order their opposite sides similarly. 

Since the measures of the three interior angles of a triangle must total , 

 has the least degree measure, so its opposite side, , is the shortest. , so by the Isosceles Triangle Theorem, their opposite sides  and  are congruent. Therefore, the correct choice is

.

One kilometer is equal to 1,000 meters, so the triangle has sides of length 100, 100, and 1,000. However,

That is, the sum of the least two sidelengths is not greater than the third. This violates the Triangle Inequality, and this triangle cannot exist.

The altitude perpendicular to the hypotenuse of a right triangle divides that triangle into two smaller triangles similar to each other and the large triangle. Therefore, the sides are in proportion. The hypotenuse of the triangle is equal to

Therefore, we can set up, and solve for  in, a proportion statement involving the shorter side and hypotenuse of the large triangle and the larger of the two smaller triangles:

This is not one of the choices.

All of the sides of a square have the same length.  

That means all four sides of this square have a length of .  

The sum of three of them would then be 

.

The area of a kite is half the product of the diagonals.

The diagonals of the kite are the height and width of the rectangle it is superimposed in, and we know that because the area of a rectangle is base times height.

Therefore our equation becomes:

.

We also know the area of the rectangle is . Substituting this value in we get the following:

Thus,, the area of the kite is .

In order to solve this problem, first observe that the red diagonal line divides the kite into two triangles that each have side lengths of  and  Notice, the hypotenuse of the interior triangle is the red diagonal. Therefore, use the Pythagorean theorem: , where  the length of the red diagonal. 

The solution is: