To find the length of the diagonal, we can consider only the triangle  and use the law of cosines to find the length of the unknown side.

The Law of Cosines:

Where  is the length of the unknown side,  and  are the lengths of the known sides, and  is the angle between  and . 

From the problem:

To find the length of the diagonal, we can consider only the triangle  and use the law of cosines to find the length of the unknown side.

The Law of Cosines:

Where  is the length of the unknown side,  and  are the lengths of the known sides, and  is the angle between  and . 

From the problem:

The formula to find the area of a parallelogram is

The base, , is given by the question.

You should recognize that  is not only the height of parallelogram , but it is also a leg of the right triangle .

Use the Pythagorean Theorem to find the length of .

Now that we have the height, multiply it by the base to find the area of the parallelogram.

The formula for the area of a parallelogram is given by the equation , where  is the base and  is the height of the parallelogram. 

The only given information is that the base is , the side is , and the base of the right triangle in the parallelogram (the triangle formed between the edge of the top base and the bottom base) is  because .

The last part of information that is required to fulfill the needs of the area formula is the parallelogram's height, . The parallelogram's height is given by the mystery side of the right triangle described in the question. In order to solve for the triangle's third side, we can use the Pythagorean Theorem, .

In this case, the unknown side is one of the legs of the triangle, so we will label it . The given side of the triangle that is part of the base we will call , and the side of the parallelogram is also the hypotenuse of the triangle, so in the Pythagorean Formula its length will be represented by . At this point, we can substitute in these values and solve for :

 , but because we're finding a length, the answer must be 4. The negative option can be negated. 

Remembering that we temporarily called  "" for the pythagorean theorem, this means that .

Now all the necessary parts for the area of a parallelogram equation are available to be used:

The length is 5 x 3 = 15 inches. Multiplied by the width of 3 inches, yields 45 in².

Rectangle

B = 2H

B = 8”

H = B/2 = 8/2 = 4”

Area = B x H = 8” X 4” = 32 in²

For a rectangle,  and , where  is the length and  is the width.

Let  be equal to the width. We know that the length is equal to "two more than twice with width."

The equation to solve for the perimeter becomes .

Now that we know the width, we can solve for the length.

Now we can find the area using .

The width of the rectangle is w, therefore the length is 3w.  The perimeter, P, can then be described as P = w + w + 3w +3w

                                                                                          40 = 8w

                                                                                          w = 5

                                                                                          width = 5, length = 3w = 15

                                                                                          A = 5*15 = 75 square inches

First, we need to find the area of the room. Because the room is rectangular, we can multiply 20 feet by 30 feet, which is 600 square feet. Next, we need to know how much space one carpet piece covers. Because the carpet pieces are also rectangular, we can multiply 4 feet by 5 feet to get 20 feet. To determine how many pieces of carpet Angela will need, we must divide the total square footage of the room (600 feet) by the square footage covered by one carpet piece (20 feet). 600 divided by 20 is 30, so Angela will need 30 carpet pieces to carpet the entire room.

the length of the rectangle is half the width, and the width is 8, so the length must be half of 8, which is 4.

The area of the rectangle can be determined from multiplying length by width, so,

4 x 8 = 32 inches squared