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Intermediate Geometry : Plane Geometry

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #42 : Rhombuses

Find the perimeter of a rhombus that has an area of 150 and a diagonal of .

Possible Answers:

Correct answer:

Explanation:

Recall the following properties of a rhombus, as shown by the figure above: The diagonals of a rhombus are perpendicular and they bisect each other. Thus, if we know the lengths of the diagonals, we can use the Pythagorean theorem to find the length of a side of the rhombus.

Recall how to find the area of a rhombus:

$\text{Area}=\frac{(\text{diagonal 1})(\text{diagonal 2})}{2}$

Since we are given the length of one diagonal and the area, we can find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(\text{Area})}{\text{diagonal 1}}$

Plug in the given values to find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(150)}{15}=20$

Now, notice that the halves of each diagonal make up a right triangle that has the side length of the rhombus as its hypotenuse.

$\text{Half diagonal 1}=\frac{15}{2}=7.5$

$\text{Half diagonal 2}=\frac{20}{2}=10$

Now, use these half values in the Pythagorean Theorem to find the length of the side of a rhombus.

$\text{Side of Rhombus}=\sqrt{7.5^2+100^2}=\sqrt{156.25}$

Finally, recall that a rhombus has four equal side lengths. To find the perimeter, multiply the length of a side by four.

$\text{Perimeter of rhombus}=4\sqrt{156.25}=50$

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Example Question #212 : Intermediate Geometry

Find the perimeter of a rhombus if it has an area of and a diagonal of .

Possible Answers:

32.81

28.84

39.67

20.36

Correct answer:

28.84

Explanation:

Recall how to find the area of a rhombus:

$\text{Area}=\frac{(\text{diagonal 1})(\text{diagonal 2})}{2}$

Since we are given the length of one diagonal and the area, we can find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(\text{Area})}{\text{diagonal 1}}$

Plug in the given values to find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(48)}{8}=12$

Now, notice that the halves of each diagonal make up a right triangle that has the side length of the rhombus as its hypotenuse.

$\text{Half diagonal 1}=\frac{8}{2}=4$

$\text{Half diagonal 2}=\frac{12}{2}=6$

Now, use these half values in the Pythagorean Theorem to find the length of the side of a rhombus.

$\text{Side of Rhombus}=\sqrt{4^2+6^2}=\sqrt{52}$

Finally, recall that a rhombus has four equal side lengths. To find the perimeter, multiply the length of a side by four.

$\text{Perimeter of rhombus}=4\sqrt{52}=28.84$

Make sure to round to places after the decimal.

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Example Question #213 : Intermediate Geometry

Find the perimeter of a rhombus if it has an area of 152 and a diagonal of .

Possible Answers:

49.68

47.01

52.33

45.26

Correct answer:

49.68

Explanation:

Recall how to find the area of a rhombus:

$\text{Area}=\frac{(\text{diagonal 1})(\text{diagonal 2})}{2}$

Since we are given the length of one diagonal and the area, we can find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(\text{Area})}{\text{diagonal 1}}$

Plug in the given values to find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(152)}{16}=19$

Now, notice that the halves of each diagonal make up a right triangle that has the side length of the rhombus as its hypotenuse.

$\text{Half diagonal 1}=\frac{16}{2}=8$

$\text{Half diagonal 2}=\frac{19}{2}=9.5$

Now, use these half values in the Pythagorean Theorem to find the length of the side of a rhombus.

$\text{Side of Rhombus}=\sqrt{8^2+9.5^2}=\sqrt{154.25}$

Finally, recall that a rhombus has four equal side lengths. To find the perimeter, multiply the length of a side by four.

$\text{Perimeter of rhombus}=4\sqrt{154.25}=49.68$

Make sure to round to places after the decimal.

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Example Question #43 : Rhombuses

Find the perimeter of a rhombus if it has an area of 187 and a diagonal of .

Possible Answers:

65.29

50.44

59.37

55.61

Correct answer:

55.61

Explanation:

Recall how to find the area of a rhombus:

$\text{Area}=\frac{(\text{diagonal 1})(\text{diagonal 2})}{2}$

Since we are given the length of one diagonal and the area, we can find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(\text{Area})}{\text{diagonal 1}}$

Plug in the given values to find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(187)}{17}=22$

Now, notice that the halves of each diagonal make up a right triangle that has the side length of the rhombus as its hypotenuse.

$\text{Half diagonal 1}=\frac{17}{2}=8.5$

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

Now, use these half values in the Pythagorean Theorem to find the length of the side of a rhombus.

$\text{Side of Rhombus}=\sqrt{8.5^2+11^2}=\sqrt{193.25}$

Finally, recall that a rhombus has four equal side lengths. To find the perimeter, multiply the length of a side by four.

$\text{Perimeter of rhombus}=4\sqrt{193.25}=55.61$

Make sure to round to places after the decimal.

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Example Question #215 : Intermediate Geometry

Find the perimeter of a rhombus if it has an area of 228 and a diagonal of .

Possible Answers:

61.22

71.85

65.03

69.81

Correct answer:

61.22

Explanation:

Recall how to find the area of a rhombus:

$\text{Area}=\frac{(\text{diagonal 1})(\text{diagonal 2})}{2}$

Since we are given the length of one diagonal and the area, we can find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(\text{Area})}{\text{diagonal 1}}$

Plug in the given values to find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(228)}{19}=24$

Now, notice that the halves of each diagonal make up a right triangle that has the side length of the rhombus as its hypotenuse.

$\text{Half diagonal 1}=\frac{19}{2}=9.5$

$\text{Half diagonal 2}=\frac{24}{2}=12$

Now, use these half values in the Pythagorean Theorem to find the length of the side of a rhombus.

$\text{Side of Rhombus}=\sqrt{9.5^2+12^2}=\sqrt{234.25}$

Finally, recall that a rhombus has four equal side lengths. To find the perimeter, multiply the length of a side by four.

$\text{Perimeter of rhombus}=4\sqrt{234.25}=61.22$

Make sure to round to places after the decimal.

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Example Question #216 : Intermediate Geometry

Find the perimeter of a rhombus if it has an area of and a diagonal of .

Possible Answers:

Correct answer:

Explanation:

Recall how to find the area of a rhombus:

$\text{Area}=\frac{(\text{diagonal 1})(\text{diagonal 2})}{2}$

Since we are given the length of one diagonal and the area, we can find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(\text{Area})}{\text{diagonal 1}}$

Plug in the given values to find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(96)}{12}=16$

Now, notice that the halves of each diagonal make up a right triangle that has the side length of the rhombus as its hypotenuse.

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

$\text{Half diagonal 2}=\frac{16}{2}=8$

Now, use these half values in the Pythagorean Theorem to find the length of the side of a rhombus.

$\text{Side of Rhombus}=\sqrt{6^2+8^2}=\sqrt{100}=10$

Finally, recall that a rhombus has four equal side lengths. To find the perimeter, multiply the length of a side by four.

$\text{Perimeter of rhombus}=4(10)=40$

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Example Question #217 : Intermediate Geometry

Find the perimeter of a rhombus that has an area of 210 and a diagonal of .

Possible Answers:

Correct answer:

Explanation:

Recall how to find the area of a rhombus:

$\text{Area}=\frac{(\text{diagonal 1})(\text{diagonal 2})}{2}$

Since we are given the length of one diagonal and the area, we can find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(\text{Area})}{\text{diagonal 1}}$

Plug in the given values to find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(210)}{20}=21$

Now, notice that the halves of each diagonal make up a right triangle that has the side length of the rhombus as its hypotenuse.

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

$\text{Half diagonal 2}=\frac{21}{2}=10.5$

Now, use these half values in the Pythagorean Theorem to find the length of the side of a rhombus.

$\text{Side of Rhombus}=\sqrt{10^2+10.5^2}=\sqrt{210.25}$

Finally, recall that a rhombus has four equal side lengths. To find the perimeter, multiply the length of a side by four.

$\text{Perimeter of rhombus}=4\sqrt{210.25}=58$

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Example Question #218 : Intermediate Geometry

Find the perimeter of a rhombus if it has an area of 1200 and a diagonal length of .

Possible Answers:

188.03

174.22

152.36

144.22

Correct answer:

144.22

Explanation:

Recall how to find the area of a rhombus:

$\text{Area}=\frac{(\text{diagonal 1})(\text{diagonal 2})}{2}$

Since we are given the length of one diagonal and the area, we can find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(\text{Area})}{\text{diagonal 1}}$

Plug in the given values to find the length of the second diagonal.

$\text{diagonal 2}=\frac{2(1200)}{40}=60$

Now, notice that the halves of each diagonal make up a right triangle that has the side length of the rhombus as its hypotenuse.

$\text{Half diagonal 1}=\frac{40}{2}=20$

$\text{Half diagonal 2}=\frac{60}{2}=30$

Now, use these half values in the Pythagorean Theorem to find the length of the side of a rhombus.

$\text{Side of Rhombus}=\sqrt{20^2+30^2}=\sqrt{1300}$

Finally, recall that a rhombus has four equal side lengths. To find the perimeter, multiply the length of a side by four.

$\text{Perimeter of rhombus}=4\sqrt{1300}=144.22$

Make sure to round to places after the decimal.

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Example Question #219 : Intermediate Geometry

Given Rhombus ABCD and diagonal $\overline{ BD}$ .

$m \angle ABD = 25 ^{\circ }$

$m \angle BCD = ?$

Possible Answers:

$110^{\circ }$

$130 ^{\circ }$

$155^{\circ }$

$105^{\circ }$

$\textup{The question cannot be answered from the information given}$

Correct answer:

$130 ^{\circ }$

Explanation:

The rhombus referenced is below:

Rhombus

A diagonal of a rhombus bisects the angles of the rhombus at its endpoints. Therefore, since $m \angle ABD = 25 ^{\circ }$ , it follows that $m \angle DBC = 25 ^{\circ }$ as well. By angle addition,

$m \angle ABC = m \angle ABD + m \angle DBC = 25^{\circ }+ 25^{\circ } = 50^{\circ }$ .

As consecutive angles of a rhombus (and, consequently, of a parallelogram), $\angle ABC$ and $\angle BDC$ are supplementary - that is, their measures total $180^{\circ }$ . Therefore,

$m \angle ABC + m \angle BCD = 180^{\circ }$

$50 ^{\circ } + m \angle BCD = 180^{\circ }$

$50 ^{\circ } + m \angle BCD - 50 ^{\circ } = 180^{\circ } - 50 ^{\circ }$

$m \angle BCD = 130 ^{\circ }$

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Example Question #1 : How To Find The Length Of The Side Of A Rhombus

If a rhombus has a base that is $\frac{4}{3}$ times greater than the height and the height of the rhombus is equal to , find the length of one side of the rhombus.

Possible Answers:

$\frac{9}{4}$

Correct answer:

Explanation:

To find the length of a side of the rhombus, multiply times $\frac{4}{3}$ .

Thus, the solution is:

$\frac{9}{1}\times\frac{4}{3}=\frac{36}{3}=12$

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Intermediate Geometry : Plane Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #42 : Rhombuses

Example Question #212 : Intermediate Geometry

Example Question #213 : Intermediate Geometry

Example Question #43 : Rhombuses

Example Question #215 : Intermediate Geometry

Example Question #216 : Intermediate Geometry

Example Question #217 : Intermediate Geometry

Example Question #218 : Intermediate Geometry

Example Question #219 : Intermediate Geometry

Example Question #1 : How To Find The Length Of The Side Of A Rhombus

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