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Intermediate Geometry : Quadrilaterals

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Example Questions

Example Question #71 : Quadrilaterals

Find the length of a side of a rhombus if it has diagonals possessing the following lengths: and .

Possible Answers:

$2\sqrt{21}$

$2\sqrt{15}$

$\sqrt{58}$

$3\sqrt{10}$

Correct answer:

$\sqrt{58}$

Explanation:

Recall that the diagonals of a rhombus bisect each other and are perpendicular to one another.

In order to find the length of the side, we can use Pythagorean's theorem. The lengths of half of the diagonals are the legs, and the length of the side of the rhombus is the hypotenuse.

First, find the lengths of half of the diagonals.

$\frac{\text{Diagonal 1}}{2}=\frac{6}{2}=3$

$\frac{\text{Diagonal 2}}{2}=\frac{14}{2}=7$

Next, substitute these half diagonals as the legs in a right triangle using the Pythagorean theorem.

$3^2+7^2=\text{side length}^2$

$\text{side length}^2=9+49=58$

$\text{side length}=\sqrt{58}$

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Example Question #72 : Quadrilaterals

Find the length of a side of a rhombus if it has diagonals possessing the following lengths: and .

Possible Answers:

$\sqrt{165}$

$2\sqrt{165}$

$\sqrt{346}$

$\sqrt{211}$

Correct answer:

$\sqrt{346}$

Explanation:

Recall that the diagonals of a rhombus bisect each other and are perpendicular to one another.

In order to find the length of the side, we can use Pythagorean's theorem. The lengths of half of the diagonals are the legs, and the length of the side of the rhombus is the hypotenuse.

First, find the lengths of half of the diagonals.

$\frac{\text{Diagonal 1}}{2}=\frac{22}{2}=11$

$\frac{\text{Diagonal 2}}{2}=\frac{30}{2}=15$

Next, substitute these half diagonals as the legs in a right triangle using the Pythagorean theorem.

$11^2+15^2=\text{side length}^2$

$\text{side length}^2=121+225=346$

$\text{side length}=\sqrt{346}$

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Example Question #73 : Quadrilaterals

Find the length of a side of a rhombus if it has diagonals possessing the following lengths: and .

Possible Answers:

$\sqrt{371}$

$\sqrt{269}$

$\sqrt{277}$

$\sqrt{259}$

Correct answer:

$\sqrt{269}$

Explanation:

Recall that the diagonals of a rhombus bisect each other and are perpendicular to one another.

In order to find the length of the side, we can use Pythagorean's theorem. The lengths of half of the diagonals are the legs, and the length of the side of the rhombus is the hypotenuse.

First, find the lengths of half of the diagonals.

$\frac{\text{Diagonal 1}}{2}=\frac{20}{2}=10$

$\frac{\text{Diagonal 2}}{2}=\frac{26}{2}=13$

Next, substitute these half diagonals as the legs in a right triangle using the Pythagorean theorem.

$10^2+13^2=\text{side length}^2$

$\text{side length}^2=100+169=269$

$\text{side length}=\sqrt{269}$

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Example Question #241 : Intermediate Geometry

Find the length of a side of a rhombus if it has diagonals possessing the following lengths: and .

Possible Answers:

$\sqrt{133}$

$\sqrt{159}$

$\sqrt{149}$

Correct answer:

$\sqrt{149}$

Explanation:

Recall that the diagonals of a rhombus bisect each other and are perpendicular to one another.

In order to find the length of the side, we can use Pythagorean's theorem. The lengths of half of the diagonals are the legs, and the length of the side of the rhombus is the hypotenuse.

First, find the lengths of half of the diagonals.

$\frac{\text{Diagonal 1}}{2}=\frac{14}{2}=7$

$\frac{\text{Diagonal 2}}{2}=\frac{20}{2}=10$

Next, substitute these half diagonals as the legs in a right triangle using the Pythagorean theorem.

$7^2+10^2=\text{side length}^2$

$\text{side length}^2=49+100=149$

$\text{side length}=\sqrt{149}$

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Example Question #74 : Quadrilaterals

Find the length of a side of a rhombus that has diagonal lengths of and .

Possible Answers:

6.71

8.96

10.92

9.22

Correct answer:

9.22

Explanation:

Recall that in a rhombus, the diagonals are not only perpendicular to each other, but also bisect one another.

Thus, we can find the lengths of half of each diagonal and use that in the Pythagorean Theorem to find the length of the side of the rhombus.

First, find the lengths of half of each diagonal.

$\text{Half Diagonal 1}=\frac{12}{2}=6$

$\text{Half Diagonal 2}=\frac{14}{2}=7$

Now, use these half diagonals as the legs of a right triangle that has the side of the rhombus as its hypotenuse.

$\text{Side of Rhombus}=\sqrt{(\text{Half Diagonal 1})^2+(\text{Half Diagonal 2})^2}$

Plug in the lengths of the half diagonals to find the length of the rhombus.

$\text{Side of Rhombus}=\sqrt{6^2+7^2}=\sqrt{85}=9.22$

Make sure to round to places after the decimal.

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Example Question #76 : Rhombuses

A garden is shaped like a rhombus. If the diagonals of the garden are feet and feet in length, in feet, what is the length of feet for a side of the garden?

Possible Answers:

12.44

8.60

15.47

5.09

Correct answer:

8.60

Explanation:

Recall that in a rhombus, the diagonals are not only perpendicular to each other, but also bisect one another.

Thus, we can find the lengths of half of each diagonal and use that in the Pythagorean Theorem to find the length of the side of the rhombus.

First, find the lengths of half of each diagonal.

$\text{Half Diagonal 1}=\frac{10}{2}=5$

$\text{Half Diagonal 2}=\frac{14}{2}=7$

Now, use these half diagonals as the legs of a right triangle that has the side of the rhombus as its hypotenuse.

$\text{Side of Rhombus}=\sqrt{(\text{Half Diagonal 1})^2+(\text{Half Diagonal 2})^2}$

Plug in the lengths of the half diagonals to find the length of the rhombus.

$\text{Side of Rhombus}=\sqrt{5^2+7^2}=\sqrt{74}=8.60$

Make sure to round to places after the decimal.

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Example Question #75 : Quadrilaterals

Find the length of a side of a rhombus that has diagonals with lengths of and .

Possible Answers:

2.69

2.10

1.72

1.51

Correct answer:

2.69

Explanation:

Recall that in a rhombus, the diagonals are not only perpendicular to each other, but also bisect one another.

Thus, we can find the lengths of half of each diagonal and use that in the Pythagorean Theorem to find the length of the side of the rhombus.

First, find the lengths of half of each diagonal.

$\text{Half Diagonal 1}=\frac{2}{2}=1$

$\text{Half Diagonal 2}=\frac{5}{2}$

Now, use these half diagonals as the legs of a right triangle that has the side of the rhombus as its hypotenuse.

$\text{Side of Rhombus}=\sqrt{(\text{Half Diagonal 1})^2+(\text{Half Diagonal 2})^2}$

Plug in the lengths of the half diagonals to find the length of the rhombus.

$\text{Side of Rhombus}=\sqrt{1^2+(\frac{5}{2})^2}=2.69$

Make sure to round to places after the decimal.

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Example Question #78 : Rhombuses

Find the length of a side of a rhombus that has diagonals with lengths and

Possible Answers:

2.11

3.61

3.89

4.10

Correct answer:

3.61

Explanation:

Recall that in a rhombus, the diagonals are not only perpendicular to each other, but also bisect one another.

Thus, we can find the lengths of half of each diagonal and use that in the Pythagorean Theorem to find the length of the side of the rhombus.

First, find the lengths of half of each diagonal.

$\text{Half Diagonal 1}=\frac{4}{2}=2$

$\text{Half Diagonal 2}=\frac{6}{2}=3$

Now, use these half diagonals as the legs of a right triangle that has the side of the rhombus as its hypotenuse.

$\text{Side of Rhombus}=\sqrt{(\text{Half Diagonal 1})^2+(\text{Half Diagonal 2})^2}$

Plug in the lengths of the half diagonals to find the length of the rhombus.

$\text{Side of Rhombus}=\sqrt{2^2+3^2}=\sqrt{13}=3.61$

Make sure to round to places after the decimal.

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Example Question #71 : Quadrilaterals

Find the length of a side of a rhombus that has diagonals with lengths of and .

Possible Answers:

15.01

14.87

14.22

16.39

Correct answer:

14.87

Explanation:

Recall that in a rhombus, the diagonals are not only perpendicular to each other, but also bisect one another.

Thus, we can find the lengths of half of each diagonal and use that in the Pythagorean Theorem to find the length of the side of the rhombus.

First, find the lengths of half of each diagonal.

$\text{Half Diagonal 1}=\frac{20}{2}=10$

$\text{Half Diagonal 2}=\frac{22}{2}=11$

Now, use these half diagonals as the legs of a right triangle that has the side of the rhombus as its hypotenuse.

$\text{Side of Rhombus}=\sqrt{(\text{Half Diagonal 1})^2+(\text{Half Diagonal 2})^2}$

Plug in the lengths of the half diagonals to find the length of the rhombus.

$\text{Side of Rhombus}=\sqrt{10^2+11^2}=\sqrt{221}=14.87$

Make sure to round to places after the decimal.

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Example Question #77 : Quadrilaterals

Find the length of a side of a rhombus that has diagonals with lengths of and .

Possible Answers:

19.24

18.44

19.05

18.51

Correct answer:

18.44

Explanation:

Recall that in a rhombus, the diagonals are not only perpendicular to each other, but also bisect one another.

Thus, we can find the lengths of half of each diagonal and use that in the Pythagorean Theorem to find the length of the side of the rhombus.

First, find the lengths of half of each diagonal.

$\text{Half Diagonal 1}=\frac{24}{2}=12$

$\text{Half Diagonal 2}=\frac{28}{2}=14$

Now, use these half diagonals as the legs of a right triangle that has the side of the rhombus as its hypotenuse.

$\text{Side of Rhombus}=\sqrt{(\text{Half Diagonal 1})^2+(\text{Half Diagonal 2})^2}$

Plug in the lengths of the half diagonals to find the length of the rhombus.

$\text{Side of Rhombus}=\sqrt{12^2+14^2}=\sqrt{340}=18.44$

Make sure to round to places after the decimal.

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Intermediate Geometry : Quadrilaterals

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #71 : Quadrilaterals

Example Question #72 : Quadrilaterals

Example Question #73 : Quadrilaterals

Example Question #241 : Intermediate Geometry

Example Question #74 : Quadrilaterals

Example Question #76 : Rhombuses

Example Question #75 : Quadrilaterals

Example Question #78 : Rhombuses

Example Question #71 : Quadrilaterals

Example Question #77 : Quadrilaterals

All Intermediate Geometry Resources

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