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Advanced Geometry : Quadrilaterals

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Example Question #1 : How To Find The Length Of The Diagonal Of A Kite

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The area of the rectangle is , what is the area of the kite?

Possible Answers:

Correct answer:

Explanation:

The area of a kite is half the product of the diagonals.

$A=\frac{p \cdot q}{2}$

The diagonals of the kite are the height and width of the rectangle it is superimposed in, and we know that because the area of a rectangle is base times height.

Therefore our equation becomes:

$A=\frac{l \cdot w}{2}$ .

We also know the area of the rectangle is $A=l \cdot w=80$ . Substituting this value in we get the following:

$A=\frac{l \cdot w}{2}=\frac{80}{2}=40$

Thus,, the area of the kite is .

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Example Question #2 : How To Find The Length Of The Diagonal Of A Kite

Given: Quadrilateral KITE such that $\overline{KI} \cong \overline{KE}$ , $\overline{TI} \cong \overline{TE}$ , $m \angle K = 60^{\circ }$ , $\angle T$ is a right angle, and diagonal $\overline{I E}$ has length 24.

Give the length of diagonal $\overline{KT}$ .

Possible Answers:

$12+ 12\sqrt{2}$

$12\sqrt{2}+ 12\sqrt{3}$

None of the other responses is correct.

$12+ 12\sqrt{3}$

Correct answer:

$12+ 12\sqrt{3}$

Explanation:

The Quadrilateral KITE is shown below with its diagonals $\overline{I E}$ and $\overline{KT}$ .

. We call the point of intersection :

Kite

The diagonals of a quadrilateral with two pairs of adjacent congruent sides - a kite - are perpendicular; also, $\overline{KT}$ bisects the $60 ^{\circ }$ and $90^{\circ }$ angles of the kite. Consequently, $\bigtriangleup KXI$ is a 30-60-90 triangle and $\bigtriangleup TXI$ is a 45-45-90 triangle. Also, the diagonal that connects the common vertices of the pairs of adjacent sides bisects the other diagonal, making the midpoint of $\overline{IE}$ . Therefore,

$IX = \frac{1}{2} \cdot IE = \frac{1}{2} \cdot 24 = 12$ .

By the 30-60-90 Theorem, since $\overline{IX}$ and $\overline{KX}$ are the short and long legs of $\bigtriangleup KXI$ ,

$KX = IX \cdot \sqrt{3}= 12 \sqrt{3}$

By the 45-45-90 Theorem, since $\overline{IX}$ and $\overline{XT}$ are the legs of a 45-45-90 Theorem,

XT = IX = 12 .

The diagonal $\overline{KT}$ has length

$KT =XT + KX= 12+ 12\sqrt{3}$ .

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Example Question #3 : How To Find The Length Of The Diagonal Of A Kite

Kite vt act

Using the kite shown above, find the length of the red (vertical) diagonal.

Possible Answers:

17.5

15.5

19.5

Correct answer:

Explanation:

In order to solve this problem, first observe that the red diagonal line divides the kite into two triangles that each have side lengths of and Notice, the hypotenuse of the interior triangle is the red diagonal. Therefore, use the Pythagorean theorem: a^2+b^2=c^2 , where the length of the red diagonal.

The solution is:

8^2+15^2=c^2

64+225=c^2

c^2=289

$c=\sqrt{289}=\sqrt{17\times 17}=17$

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Example Question #4 : How To Find The Length Of The Diagonal Of A Kite

A kite has two perpendicular interior diagonals. One diagonal is twice the length of the other diagonal. The total area of the kite is $196\textup{ units}^{2}$ . Find the length of each interior diagonal.

Possible Answers:

$7\textup{ and }14$

$7^2\textup{ and }15$

$15\textup{ and }45$

$15\textup{ and }30$

$14\textup{ and }28$

Correct answer:

$14\textup{ and }28$

Explanation:

To solve this problem, apply the formula for finding the area of a kite:

$Area=\frac{diagonalA\times diagonalB}{2}$

However, in this problem the question only provides information regarding the exact area. The lengths of the diagonals are represented as a ratio, where
diagonalA:diagonalB=1:2

Therefore, it is necessary to plug the provided information into the area formula. Diagonal is represented by and diagonal B=2 .

The solution is:

$196=\frac{x\times 2x}{2}$

$196\times2=x\times 2x$

392=2x^2

$x^2=\frac{392}{2}=196$

$x=\sqrt{196}=14$

Thus, if x=14 , then diagonal must equal 2(14)=28

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Example Question #1 : How To Find The Length Of The Diagonal Of A Kite

A kite has two perpendicular interior diagonals. One diagonal has a measurement of and the area of the kite is $60\textup{ units}^{2}$ . Find the length of the other interior diagonal.

Possible Answers:

7.5

5.5

$\sqrt{15}$

Correct answer:

Explanation:

This problem can be solved by applying the area formula:

Since this question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal.

Thus the solution is:

$60=\frac{8\times diagonal B}{2}$

$60\times2=8\times diagonalB$

120=8(diagonalB)

$diagonal B=\frac{120}{8}=15$

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Example Question #6 : How To Find The Length Of The Diagonal Of A Kite

A kite has two perpendicular interior diagonals. One diagonal has a measurement of and the area of the kite is $45\textup{ units}^{2}$ . Find the length of the other interior diagonal.

Possible Answers:

5.5

8.5

Correct answer:

Explanation:

This problem can be solved by applying the area formula:

Since this question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal.

Thus the solution is:

$45=\frac{18\times diagonal B}{2}$

$45\times2=18\times diagonalB$

90=18(diagonalB)

$diagonal B=\frac{90}{18}=5$

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Example Question #1 : Kites

A kite has two perpendicular interior diagonals. One diagonal has a measurement of 250 and the area of the kite is $6\textup{,}250\textup{ units}^{2}$ . Find the sum of the two perpendicular interior diagonals.

Possible Answers:

450

200

300

$2\textup{,}000$

Correct answer:

300

Explanation:

First find the length of the missing diagonal before you can find the sum of the two perpendicular diagonals.

To find the missing diagonal, apply the area formula:

This question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal.

$6,250=\frac{250\times diagonal B}{2}$

$6250\times2=250\times diagonalB$

12,500=250(diagonalB)

$diagonal B=\frac{12,500}{250}=50$

Therefore, the sum of the two diagonals is:

250+50=300

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Example Question #8 : How To Find The Length Of The Diagonal Of A Kite

A kite has two perpendicular interior diagonals. One diagonal has a measurement of and the area of the kite is $28\textup{ units}^{2}$ . Find the sum of the two perpendicular interior diagonals.

Possible Answers:

Correct answer:

Explanation:

You must find the length of the missing diagonal before you can find the sum of the two perpendicular diagonals.

To find the missing diagonal, apply the area formula:

This question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal.

$28=\frac{4\times diagonal B}{2}$

$28\times2=4\times diagonalB$

56=4(diagonalB)

$diagonal B=\frac{56}{4}=14$

Therefore, the sum of the two diagonals is:

14+4=18

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Example Question #9 : How To Find The Length Of The Diagonal Of A Kite

Kite vt act

The area of the kite shown above is $125\textup{ units}^{2}$ and the red diagonal has a length of . Find the length of the black (horizontal) diagonal.

Possible Answers:

Correct answer:

Explanation:

To find the length of the black diagonal apply the area formula:

Since this question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal.

Thus the solution is:

$125=\frac{25\times diagonal B}{2}$

$125\times2=25\times diagonalB$

250=25(diagonalB)

$diagonal B=\frac{250}{25}=10$

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Example Question #2 : Kites

A kite has two perpendicular interior diagonals. One diagonal has a measurement of $22\textup{mm}$ and the area of the kite is $297\textup{mm}^{2}$ . Find the length of the other interior diagonal.

Possible Answers:

$25\textup{mm}$

$37\textup{mm}$

$35\textup{mm}$

$27\textup{mm}$

$19\textup{mm}$

Correct answer:

$27\textup{mm}$

Explanation:

This problem can be solved by applying the area formula:

Since this question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal.

Thus the solution is:

$297=\frac{22\times diagonal B}{2}$

$297\times2=22\times diagonalB$

594=22(diagonalB)

$diagonal B=\frac{594}{22}=27$

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Advanced Geometry : Quadrilaterals

Study concepts, example questions & explanations for Advanced Geometry

All Advanced Geometry Resources

Example Questions

Example Question #1 : How To Find The Length Of The Diagonal Of A Kite

Example Question #2 : How To Find The Length Of The Diagonal Of A Kite

Example Question #3 : How To Find The Length Of The Diagonal Of A Kite

Example Question #4 : How To Find The Length Of The Diagonal Of A Kite

Example Question #1 : How To Find The Length Of The Diagonal Of A Kite

Example Question #6 : How To Find The Length Of The Diagonal Of A Kite

Example Question #1 : Kites

Example Question #8 : How To Find The Length Of The Diagonal Of A Kite

Example Question #9 : How To Find The Length Of The Diagonal Of A Kite

Example Question #2 : Kites

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