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ACT Math : Quadrilaterals

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Example Question #3 : How To Find The Length Of The Diagonal Of A Parallelogram

Parallelogram_13

ABCD is a parallelogram. Find the length of diagonal $\overline{AC}$ .

Possible Answers:

108

136

112

102

Correct answer:

112

Explanation:

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

The Law of Cosines:

$c^2 = a^2 + b^2 - 2ab\cos \left ( C\right )$

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:

$\overline{AC}^2 = \overline{CD}^2 + \overline{DA}^2 - 2\left ( \overline{CD} \right )\left (\overline{DA} \right )\cos \left ( \measuredangle D\right )$

$\overline{AC}^2 = 35^2 + 82 ^2 - 2\cdot35\cdot82\cdot \cos \left ( 143^{\circ}\right )$

$\overline{AC}^2 = 12553$

$\overline{AC} = 112$

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Example Question #4 : How To Find The Length Of The Diagonal Of A Parallelogram

Parallelogram_15

ABCD is a parallelogram. Find the length of diagonal $\overline{AC}$ .

Possible Answers:

49.2

41.8

56.6

59.1

28.3

Correct answer:

49.2

Explanation:

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

The Law of Cosines:

$c^2 = a^2 + b^2 - 2ab\cos \left ( C\right )$

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:

$\overline{AC}^2 = \overline{CD}^2 + \overline{DA}^2 - 2\left ( \overline{CD} \right )\left (\overline{DA} \right )\cos \left ( \measuredangle D\right )$

$\overline{AC}^2 = 12^2 + 40 ^2 - 2\cdot12\cdot40\cdot \cos \left ( 135^{\circ}\right )$

$\overline{AC}^2 = 2423$

$\overline{AC} = 49.2$

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

Parallelogram_16

ABCD is a parallelogram. Find the length of diagonal $\overline{AC}$ .

Possible Answers:

22.4

26.5

40.0

44.8

20.0

Correct answer:

26.5

Explanation:

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

The Law of Cosines:

$c^2 = a^2 + b^2 - 2ab\cos \left ( C\right )$

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:

$\overline{AC}^2 = \overline{CD}^2 + \overline{DA}^2 - 2\left ( \overline{CD} \right )\left (\overline{DA} \right )\cos \left ( \measuredangle D\right )$

$\overline{AC}^2 = 10^2 + 20 ^2 - 2\cdot10\cdot20\cdot \cos \left ( 120^{\circ}\right )$

$\overline{AC}^2 = 700$

$\overline{AC} = 26.5$

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

Parallelogram_14

ABCD is a parallelogram. Find the length of diagonal. $\overline{BD}$ .

Possible Answers:

41.8

56.6

32.6

29.5

26.0

Correct answer:

32.6

Explanation:

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

The Law of Cosines:

$c^2 = a^2 + b^2 - 2ab\cos \left ( C\right )$

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:

$\overline{BD}^2 = \overline{AB}^2 + \overline{DA}^2 - 2\left ( \overline{AB} \right )\left (\overline{DA} \right )\cos \left ( \measuredangle A\right )$

$\overline{BD}^2 = 12^2 + 40 ^2 - 2\cdot12\cdot40\cdot \cos \left ( 45^{\circ}\right )$

$\overline{BD}^2 = 1065$

$\overline{BD} = 32.6$

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

Parallelogram_17

ABCD is a parallelogram. Find the length of diagonal $\overline{BD}$ .

Possible Answers:

$\sqrt{500}$

$\sqrt{100}$

$\sqrt{400}$

$\sqrt{300}$

$\sqrt{200}$

Correct answer:

$\sqrt{300}$

Explanation:

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

The Law of Cosines:

$c^2 = a^2 + b^2 - 2ab\cos \left ( C\right )$

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:

$\overline{BD}^2 = \overline{AB}^2 + \overline{DA}^2 - 2\left ( \overline{AB} \right )\left (\overline{DA} \right )\cos \left ( \measuredangle A\right )$

$\overline{BD}^2 = 10^2 + 20 ^2 - 2\cdot10\cdot20\cdot \cos \left ( 60^{\circ}\right )$

$\overline{BD}^2 = 300$

$\overline{BD} = \sqrt{300}$

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

In parallelogram ABCD , 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 ABCD .

Para

Possible Answers:

240

124

144

288

Correct answer:

288

Explanation:

The formula to find the area of a parallelogram is

$base \times height$

The base, , is given by the question.

DE+EC=DC

12+6=18

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

Use the Pythagorean Theorem to find the length of .

${AE}^2+{DE}^2={AD}^2$

${AE}^2+12^2=20^$

${AE}^2=256$

AE=16

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

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

A parallelogram has a base of $12\:cm$ and its side is $5\:cm$ 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.

Possible Answers:

$36\:cm^2$

$48\:cm^2$

$30\:cm^2$

$60\:cm^2$

$24\:cm^2$

Correct answer:

$48\:cm^2$

Explanation:

The formula for the area of a parallelogram is given by the equation $A =b \cdot h$ , where is the base and is the height of the parallelogram.

The only given information is that the base is $12\:cm$ , the side is $5\:cm$ , 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 $3\:cm$ because $12\:cm\div4=3\:cm$ .

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, a^2 + b^2 =c^2 .

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 :

a^2 + 3^2=5^2

a^2 + 9 = 25

a^2 = 25 - 9

a^2 = 16

$a=\sqrt{16}$

$a=\pm 4$ , 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 $h=4\:cm$ .

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

$A = (12\:cm)(4\:cm)$

$A = 48\:cm^2$

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

The length of a rectangle is 5 times its width. Its width is 3 inches long. What is the area of the rectangle in square inches?

Possible Answers:

Correct answer: 45

Explanation:

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

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

A rectangle’s base is twice its height. If the base is 8” long, what is the area of the rectangle?

Possible Answers:

24 in²

32 in²

64 in²

16 in²

12 in²

Correct answer:

32 in²

Explanation:

Rectangle

B = 2H

B = 8”

H = B/2 = 8/2 = 4”

Area = B x H = 8” X 4” = 32 in²

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

The length of a rectangle is two more than twice the width. The perimeter is 58ft. What is the area of the rectangle?

Possible Answers:

$168 \; ft^{2}$

$198 \; ft^{2}$

$154 \; ft^{2}$

$138 \; ft^{2}$

$180 \; ft^{2}$

Correct answer:

$180 \; ft^{2}$

Explanation:

For a rectangle, P = 2l + 2w and A = lw , 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."

l=2w+2

The equation to solve for the perimeter becomes 58 = 2(2w + 2) + 2w .

58 = 4w + 4+2w=6w+4

54=6w

9=w

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

l=2w + 2 = 20

Now we can find the area using A = lw .

A=(20ft)(9ft)=180ft^2

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ACT Math : Quadrilaterals

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #3 : How To Find The Length Of The Diagonal Of A Parallelogram

Example Question #4 : How To Find The Length Of The Diagonal Of A Parallelogram

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

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

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

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

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

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

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

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

All ACT Math Resources

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