The answer is independent of the sidelengths of the rectangle, so to ease calculations, we will arbitrarily assign to Square  sidelength 4 inches. 

The shaded region can be divided by a perpendicular segment from  to , then another from  to that segment, as follows:

By the fact that rectangles have opposite sides that are congruent, and by the fact that  is the midpoint of and  is the midpoint of , the shaded region is a composite of:

Square , which has sides of length 2 and therefore area ;

Right triangle , which has two legs of length 2 and, thus, area ; and,

Right triangle , which has legs of length 4 and 2 and, thus, area .

The area of the shaded region is therefore .

Square  has area , so the shaded region is

of the square.

This triangle is isosceles by the converse of the Isosceles Triangle Theorem. The altitude of the triangle divides it into two triangles that are both right and that are congruent, as follows:

Each triangle is a  triangle. The altitude is the short leg of each triangle, and, therefore, its length is half that of the common hypotenuse, or 6. The long leg of each right triangle has length  times that of the short leg, or , and the base of the entire large triangle is twice this, or . 

The area of the triangle is half the product of the height and the base:

The area of a triangle is half the product of its baselength and its height. 

To find the area of , we can use the lengths of the legs  and :

To find the area of , we can use the hypotenuse , the length of which is 30, and the altitude  perpendicular to it:

In terms of area,  is

of .

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; the blue triangle has legs 20 and , where  can be calculated with the following proportion:

The legs of the blue triangle are 20 and 40; half their product is the area:

To find the area of a triangle, we will use the following formula:

where b  is the base and h is the height of the triangle.

Now, we know the base of the triangle is 7in. We also know the height of the triangle is two times the base. Therefore, the height is 14in.  So, we can substitute. We get

To find the area of a triangle, we will use the following formula:

where b is the base and h is the height of the triangle. 

Now, we know the base of the triangle is 8in. We also know the height of the triangle is half the base. Therefore, the height is 4in. So, we can substitute. We get

Write the formula for the area of a triangle.

Substitute the base and height.

Simplify the fractions.

The answer is:  

To find the area of a triangle, we will use the following formula:

where b  is the base and h is the height of the triangle.

Now, we know the base of the triangle is 6cm. We also know the height is three times the base. Therefore, the height is 18cm. So, we substitute. We get

To find the area of a triangle, we will use the following formula:

where b  is the base and h is the height of the triangle.

Now, we know the base of the triangle is 10in. We know the height of the triangle is 9in. So, we can substitute. We get

To find the area of a triangle, we will use the following formula:

where b  is the base and h is the height of the triangle.

We know the base of the triangle is 8cm.

We know the height of the triangle is 12cm.

Now, we can substitute. We get