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Intermediate Geometry : Parallelograms

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Example Question #1 : Parallelograms

The length of a parallelogram is cm and the width is cm. One of its diagonals measures cm. Find the length of the other diagonal.

Possible Answers:

$3\sqrt{3}cm$

$4\sqrt{38}cm$

$\sqrt{142}cm$

$2\sqrt{38}cm$

None of the other answers.

Correct answer:

$\sqrt{142}cm$

Explanation:

The formula for the relationship between diagonals and sides of a parallologram is

$(d_{1})^{2} + (d_{2})^{2} = 2a^{2} + 2b^{2}$ ,

where $d_{1}$ represents one diagonal,

$d_{2}$ represents the other diagonal,

represents a side, and

represents the adjoining side.

So, in this problem, substitute the known values and solve for the missing diagonal.

$14^{2} + (d_{2})^{2} = 2(12)^{2} + 2(5)^{2} \rightarrow 196 + d^{2} = 288 + 50$

$\rightarrow d^{2} = 142 \rightarrow d = \sqrt{142}$

So, the missing diagonal is $\sqrt{142}$ cm.

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Example Question #2 : Parallelograms

In the parallogram above, find the length of the labeled diagonal.

Possible Answers:

None of the other answers.

Correct answer:

Explanation:

In a parallogram, diagonals bisect one another, thus you can set the two segments that are labeled in the picture equal to one another, then solve for .

So,

$2x - 2 = x + 12 \rightarrow x = 14$ .

If x = 14 , then you can substitute 14 into each labeled segment, to get a total of 52.

2(14)-2+14+12

28-2+14+12=52

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Example Question #3 : Parallelograms

In the parallogram below, find the length of the labeled diagonal.

Parallelogram2

Possible Answers:

None of the other answers.

Correct answer:

Explanation:

In a parallelogram, the diagonals bisect one another, so you can set the labeled segments equal to each other and then solve for .

$x + 14 = 5x - 10 \rightarrow 24 = 4x \rightarrow 6 = x$ .

If x = 6 , then you substitute 6 into each labeled segment, to get a total of 40.

6+14+5(6)-10

20+30-10=40

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Example Question #4 : Parallelograms

Para3

In the parallelogram above, find the length of the labeled diagonal.

Possible Answers:

11.2

None of the other answers.

5.6

4.8

9.6

Correct answer:

11.2

Explanation:

In a parallelogram, the diagonals bisect each other, so you can set the labeled segments equal to one another and then solve for .

$20 - 3x = 2x - 4 \rightarrow 24 = 5x \rightarrow 4.8 = x$ .

Then, substitute 4.8 for in each labeled segment to get a total of 11.2 for the diagonal length.

20-3(4.8)+2(4.8)-4

20-14.4+9.6-4=11.2

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Example Question #5 : Parallelograms

Suppose a square has an area of 6. What is the diagonal of the parallelogram?

Possible Answers:

$6\sqrt2$

$\frac{3\sqrt2}{2}$

$3\sqrt3$

$3\sqrt2$

$2\sqrt3$

Correct answer:

$2\sqrt3$

Explanation:

Write the formula to find the side of the square given the area.

A=s^2

Find the side.

6=s^2

$s=\sqrt6$

The diagonal of the square can be solved by using the Pythagorean Theorem.

a^2+b^2=c^2

$a=b=s=\sqrt6$

Substitute and solve for the diagonal, .

$(\sqrt6)^2+(\sqrt6)^2 =c^2$

6+6=c^2

c^2=12

$c=\sqrt{12}=2\sqrt3$

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Example Question #6 : Parallelograms

If the side length of a square is , what is the diagonal of the square?

Possible Answers:

$9a\sqrt2$

$a^2\sqrt6$

$3a^2\sqrt2$

$3a\sqrt2$

$a\sqrt6$

Correct answer:

$3a\sqrt2$

Explanation:

Write the diagonal formula for a square.

$d=s\sqrt2$

Substitute the side length and reduce.

$d=3a\sqrt2$

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Example Question #7 : Parallelograms

Parallelogram ABCD has diagonals $\overline{AC}$ and $\overline{BD}$ . AC = 12 and BD = 6 .

True, false, or undetermined: Parallelogram ABCD is a rectangle.

Possible Answers:

True

Undetermined

False

Correct answer:

False

Explanation:

One characteristic of a rectangle is that its diagonals are congruent. Since the diagonals of Parallelogram ABCD are of different lengths, it cannot be a rectangle.

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Example Question #8 : Parallelograms

Parallelogram ABCD has diagonals $\overline{AC}$ and $\overline{BD}$ . AC = 12 and BD = 6 .

True, false, or undetermined: Parallelogram ABCD is a rhombus.

Possible Answers:

False

Undetermined

True

Correct answer:

Undetermined

Explanation:

One characteristic of a rhombus is that its diagonals are perpendicular; no restrictions exist as to their lengths. Whether or not the diagonals are perpendicular is not stated, so the figure may or may not be a rhombus.

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Example Question #9 : Parallelograms

Parallelogram_custom_2

Find the perimeter of the parallelogram shown above.

Possible Answers:

$\small 48$

$\small 19$

$\small 26$

$\small 40$

Correct answer:

$\small 26$

Explanation:

In order to find the perimeter of this parallelogram, apply the formula:
$\small Perimeter=2(base+side)$ .

The solution is:

$\small P=2(8+5)$

$\small \small P=2(13)=26$

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Example Question #10 : Parallelograms

Parallelogram_custom_3

Find the perimeter of the parallelogram shown above.

Possible Answers:

$\small 112$

$\small 224$

$\small 110$

$\small 248$

Correct answer:

$\small 224$

Explanation:

To find the perimeter of this parallelogram, first find the length of the side: $\small h\times\frac{1}{3}$ .

Since, $\small h=48$ , the side must be $\small 48\div 3=16$ .

Then apply the formula:

$\small p=2(base+side)$

$\small p=2(96+16)$

$\small p=2(112)=224$

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Intermediate Geometry : Parallelograms

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #1 : Parallelograms

Example Question #2 : Parallelograms

Example Question #3 : Parallelograms

Example Question #4 : Parallelograms

Example Question #5 : Parallelograms

Example Question #6 : Parallelograms

Example Question #7 : Parallelograms

Example Question #8 : Parallelograms

Example Question #9 : Parallelograms

Example Question #10 : Parallelograms

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