ACT Math : Parallelograms

Example Questions

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Example Question #5 : How To Find The Length Of The Diagonal Of A Parallelogram

is a parallelogram. Find the length of diagonal .

Explanation:

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:

Example Question #6 : How To Find The Length Of The Diagonal Of A Parallelogram

is a parallelogram. Find the length of diagonal. .

Explanation:

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:

Example Question #7 : How To Find The Length Of The Diagonal Of A Parallelogram

is a parallelogram. Find the length of diagonal .

Explanation:

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:

Example Question #1 : How To Find The Area Of A Parallelogram

In parallelogram , the length of  is  units, the length of  is  units, and the length of  is  units.  is perpendicular fo . Find the area, in square units, of .

Explanation:

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.

Example Question #2 : How To Find The Area Of A Parallelogram

A parallelogram has a base of  and its side is  long. A line is drawn to connect the edge of the top base with the bottom base. The line is perpendicular to the bottom base, and the base of this triangle is one-fourth the length of the bottom base. Find the area of the parallelogram.

Explanation:

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:

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