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Pre-Algebra : Geometry

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Example Questions

Example Question #2 : Area Of A Parallelogram

Parallelogram

Refer to the above figure, which shows Parallelogram ABCD . You are given that $\overline{AX} \perp \overline{BC}$ and $\overline{AY} \perp \overline{CD}$ .

If you know the length of $\overline{AD}$ , then, of the following segments, choose the one whose length, if known, will allow us to calculate the area of Parallelogram ABCD .

Possible Answers:

$\overline{AX}$

$\overline{AB}$

$\overline{BC}$

$\overline{CD}$

$\overline{AY}$

Correct answer:

$\overline{AX}$

Explanation:

The area of a parallelogram is the product of the length of any one side, or its base, and the perpendicular distance to the opposite side, or its height. If we know $\overline{AD}$ , then we also know $\overline{BC}$ , which is of the same length. We can take $\overline{BC}$ to be the base, and the segement perpendicular to it, $\overline{AX}$ , as the altitude. Therefore, $\overline{AX}$ is the segment whose length we need to know.

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Example Question #3 : Area Of A Parallelogram

Parallelogram

Note: Figure NOT drawn to scale.

Refer to the above figure, which shows Parallelogram ABCD . You are given that $\overline{AX} \perp \overline{BC}$ and $\overline{AY} \perp \overline{CD}$ .

AD = 100 , AB = 120 . AY = 80 .

Evaluate .

Possible Answers:

AX = 140

$AX = 66\frac{2}{3}$

AX = 150

AX = 60

AX = 96

Correct answer:

AX = 96

Explanation:

The area of a parallelogram is the product of the length of any one side, or its base, and the length of a segment perpendicular to that side, or its height.

One way to find the area is to multiply the length of side $\overline{CD}$ by its corresponding altitude, $\overline{AY}$ . Since CD = AB = 120 and AY = 80 ,

$A = (CD)(AY) = 120 \cdot 80 = 9,600$ .

Another way to find the area is to multiply the length of side $\overline{BC}$ by its corresponding altitude, $\overline{AX}$ . Since BC = AD= 100 and the area is 9,600, we set up this equation and solve for :

A = (BC)(AX)

$100 \cdot AX =9,600$

AX = 96

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Example Question #3 : Area Of A Parallelogram

A parallelogram has the base length of and the altitude of 2t-1 . Give the area of the parallelogram.

Possible Answers:

2t^2+t

2t^2-t

t^2+t

t^2

2t^2

Correct answer:

2t^2-t

Explanation:

The area of a parallelogram is given by:

Area=ba

Where is the base length and is the corresponding altitude. So we can write:

Area=t(2t-1)=2t^2-t

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Example Question #4 : Area Of A Parallelogram

A parallelogram has a base length of which is 3 times longer than its corresponding altitude. The area of the parallelogram is 12 square inches. Give the .

Possible Answers:

$6\sqrt{3}$

$6\sqrt{2}$

Correct answer:

Explanation:

Base length is so the corresponding altitude is $\frac{t}{3}$ .

The area of a parallelogram is given by:

Area=ba

Where:

is the length of any base
is the corresponding altitude

So we can write:

$12=t\times \frac{t}{3}\Rightarrow t\times t=12\times 3\Rightarrow t^2=36\Rightarrow t=6$

$t\times t=12\times 3$

$t^{2}=36$

t=6

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Example Question #4 : Area Of A Parallelogram

The length of the shorter diagonal of a rhombus is 40% that of the longer diagonal. The area of the rhombus is . Give the length of the longer diagonal in terms of .

Possible Answers:

$\frac{1}{5}\sqrt{K}$

$K\sqrt{\frac{5}{2}}$

$\sqrt{\frac{5K}{2}}$

$\sqrt{5K}$

$K\sqrt{\frac{2}{5}}$

Correct answer:

$\sqrt{5K}$

Explanation:

Let be the length of the longer diagonal. Then the shorter diagonal has length 40% of this. Since 40% is equal to $\frac{40}{100 } = \frac{40 \div 20 }{100 \div 20 } = \frac{2}{5}$ , 40% of is equal to $\frac{2}{5}D$ .

The area of a rhombus is half the product of the lengths of its diagonals, so we can set up, and solve for , in the equation:

$\frac{1}{2} \cdot \frac{2}{5}D \cdot D = K$

$\frac{1}{5}D^{2}= K$

$5\cdot \frac{1}{5}D^{2}=5\cdot K$

$D^{2}=5K$

$D =\sqrt{5K}$

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Example Question #4 : Area Of A Parallelogram

The length of the shorter diagonal of a rhombus is two-thirds that of the longer diagonal. The area of the rhombus is square yards. Give the length of the longer diagonal, in inches, in terms of .

Possible Answers:

$\sqrt{ 3Q}$

$3\sqrt{ 3Q}$

$36\sqrt{ 3Q }$

$4\sqrt{ 3Q}$

$12\sqrt{ 3Q}$

Correct answer:

$36\sqrt{ 3Q }$

Explanation:

Let be the length of the longer diagonal in yards. Then the shorter diagonal has length two-thirds of this, or $\frac{2}{3}D$ .

The area of a rhombus is half the product of the lengths of its diagonals, so we can set up the following equation and solve for :

$\frac{1}{2} \cdot \frac{2}{3}D \cdot D = Q$

$\frac{1}{3}D ^{2} = Q$

$3\cdot \frac{1}{3}D ^{2} = 3 \cdot Q$

$D ^{2} = 3Q$

$D = \sqrt{3Q}$

To convert yards to inches, multiply by 36:

$\sqrt{ 3Q } \times 36 = 36\sqrt{ 3Q }$

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Example Question #4 : Area Of A Parallelogram

The longer diagonal of a rhombus is 20% longer than the shorter diagonal; the rhombus has area . Give the length of the shorter diagonal in terms of .

Possible Answers:

$\sqrt{5N}$

$\sqrt{ \frac{N}{5}}$

$\sqrt{15N}$

$\sqrt{ \frac{5N}{3}}$

$\sqrt{ \frac{3N}{5}}$

Correct answer:

$\sqrt{ \frac{5N}{3}}$

Explanation:

Let be the length of the shorter diagonal. If the longer diagonal is 20% longer, then it measures 120% of the length of the shorter diagonal; this is

$\frac{120}{100} = \frac{120 \div 20}{100 \div 20} = \frac{6}{5}$

of , or $\frac{6}{5}D$ .

The area of a rhombus is half the product of the lengths of its diagonals, so we can set up an equation and solve for :

$\frac{1}{2} \cdot \frac{6}{5}D \cdot D = N$

$\frac{3}{5}D^{2} = N$

$\frac{5}{3} \cdot \frac{3}{5}D^{2} = \frac{5}{3} \cdot N$

$D^{2} = \frac{5N}{3}$

$D =\sqrt{ \frac{5N}{3}}$

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Example Question #2 : Parallelograms

Find the area:

Question_5

Possible Answers:

$\small 12$

$\small 32$

$\small 16$

$\small 24$

Correct answer:

$\small 24$

Explanation:

The area of a parallelogram can be determined using the following equation:

$\small A=bh$

Therefore,

$\small A=8\times3=24$

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Example Question #3 : Area Of A Parallelogram

Find the area of a parallelogram with a base of length 8 and a height of length 6.

Possible Answers:

A=48

A=24

A=28

A=36

A=7

Correct answer:

A=48

Explanation:

The area of a parallelogram is A=bh=(base)(height)

A=(6)(8)=48

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Example Question #51 : Area

What is the area of a parallelogram with a base of and a height of ?

Possible Answers:

112

Correct answer:

Explanation:

Write the formula for the area of a parallelogram.

A=bh

Substitute the dimensions.

$A=7\times 8 = 56$

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Pre-Algebra : Geometry

Study concepts, example questions & explanations for Pre-Algebra

All Pre-Algebra Resources

Example Questions

Example Question #2 : Area Of A Parallelogram

Example Question #3 : Area Of A Parallelogram

Example Question #3 : Area Of A Parallelogram

Example Question #4 : Area Of A Parallelogram

Example Question #4 : Area Of A Parallelogram

Example Question #4 : Area Of A Parallelogram

Example Question #4 : Area Of A Parallelogram

Example Question #2 : Parallelograms

Example Question #3 : Area Of A Parallelogram

Example Question #51 : Area

All Pre-Algebra Resources

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