Parallelograms - ACT Math

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Question

If a rectangular plot measures by , what is the length of the diagonal of the plot, in feet?

Answer

To answer this question, we must find the diagonal of a rectangle that is by . 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 .

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Question

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

Answer

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:

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

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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Question

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

Answer

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:

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

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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Question

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

Answer

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:

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

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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Question

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

Answer

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:

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

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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Question

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

Answer

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:

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

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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Question

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

Answer

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:

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

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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Question

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 .

Answer

The formula to find the area of a parallelogram is

$base \times height$

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 .

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

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

${AE}^2=256$

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

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Question

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.

Answer

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 , 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 $12:cm\div4=3:cm$ .

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, .

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 :

$a=\sqrt{16}$

$a=\pm 4$ , 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 .

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

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Question

If a rectangular plot measures by , what is the length of the diagonal of the plot, in feet?

Answer

To answer this question, we must find the diagonal of a rectangle that is by . 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 .

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Question

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

Answer

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:

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

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:

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

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

$\overline{BD}^2 = 3365$

$\overline{BD} = 58$

Compare your answer with the correct one above

Question

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

Answer

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:

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

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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Question

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

Answer

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:

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

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:

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

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

$\overline{BD}^2 = 1065$

$\overline{BD} = 32.6$

Compare your answer with the correct one above

Question

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

Answer

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:

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

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:

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

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

$\overline{AC}^2 = 2423$

$\overline{AC} = 49.2$

Compare your answer with the correct one above

Question

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

Answer

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:

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

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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Question

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

Answer

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:

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

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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Question

In parallelogram , $\overline{AB} = 10$ and $\measuredangle A = 60^{\circ}$ . Find $\overline{CD}$ .

Answer

In a parallelogram, opposite sides are congruent. Thus,

$\overline{AB} = \overline{CD}$

$10 = \overline{CD}$

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Question

In parallelogram , $\overline{BC} = 30$ and $\measuredangle B = 120^{\circ}$ . Find $\overline{AD}$ .

Answer

In a parallelogram, opposite sides are congruent.

$\overline{BC} = \overline{AD}$

$30 = \overline{AD}$

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Question

Parallelogram has an area of . If , find $\overline{BC}$ .

Answer

The area of a parallelogram is given by:

In this problem, the height is given as and the area is . Both $\overline{AD}$ and $\overline{BC}$ are bases.

$375 = \left ( 15\right )\left ( \overline{BC} \right )$

$\overline{BC} = 25$

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Question

Find the length of the base of a parallelogram with a height of and an area of .

Answer

The formula for the area of a parallelogram is:

By plugging in the given values, we get:

$204 = b\cdot12$

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