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ACT Math : How to find the length of the diagonal of a parallelogram

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

If a rectangular plot measures $8 \:ft$ by $6\:ft$ , what is the length of the diagonal of the plot, in feet?

Possible Answers:

$10\:ft$

$9\:ft$

$7\:ft$

$13\:ft$

$11\:ft$

Correct answer:

$10\:ft$

Explanation:

To answer this question, we must find the diagonal of a rectangle that is $8\: ft$ by $6\:ft$ . Because a rectangle is made up of right angles, the diagonal of a rectangle creates a right triangle with two of the sides.

Because a right triangle is formed by the diagonal, we can use the Pythagorean Theorem, which is:

$a^{2}+b^{2}=c^{2}$

and each represent a different leg of the triangle and represents the length of the hypotenuse, which in this case is the same as the diagonal length.

We can then plug in our known values and solve for $c^{2}$

$a^{2}+b^{2}=c^{2}\rightarrow 8^{2}+6^{2}=c^{2}\rightarrow 64+36=c^{2}\rightarrow 100=c^{2}$

We now must take the square root of each side so that we can solve for

$\sqrt{100}=\sqrt{c^{2}}\rightarrow 10=c$

Therefore, the diagonal of the rectangle is $10\:ft$ .

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

Parallelogram_12

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

Possible Answers:

17.5

Correct answer:

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:

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

$\overline{BD}^2 = 35^2 + 82 ^2 - 2\cdot35\cdot82\cdot \cos \left ( 37^{\circ}\right )$

$\overline{BD}^2 = 3365$

$\overline{BD} = 58$

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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

102

136

112

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 #2 : 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:

28.3

49.2

56.6

59.1

41.8

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 #1 : 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

44.8

40.0

20.0

26.5

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 #4 : 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:

26.0

29.5

56.6

32.6

41.8

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 #1 : 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:

$\sqrt{300}$

$\sqrt{100}$

$\sqrt{200}$

$\sqrt{400}$

$\sqrt{500}$

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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ACT Math : How to find the length of the diagonal of a parallelogram

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

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

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

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

Example Question #2 : 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 #4 : 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

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