GMAT Math : DSQ: Calculating the length of a diagonal of a polygon

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Example Question #1 : Dsq: Calculating The Length Of A Diagonal Of A Polygon

Hexagon_44

The hexagon in the above diagram is regular. If $\overline{AB}$ has length 10, which of the following expressions is equal to the length of $\overline{AC}$ ?

Possible Answers:

$10\sqrt{6}$

$10\sqrt{3}$

$5\sqrt{6}$

$10\sqrt{2}$

Correct answer:

$10\sqrt{3}$

Explanation:

The answer can be seen more easily by constructing the altitude of $\bigtriangleup ABC$ from , as seen below:

Untitled

Each interior angle of a hexagon measures $120^{\circ }$ ,and the altitude also bisects $\angle ABC$ , the vertex angle of isosceles $\bigtriangleup ABC$ . $\bigtriangleup ABM$ is easily proved to be a 30-60-90 triangle, so by the 30-60-90 Triangle Theorem,

$BM = \frac{1}{2} \cdot AB = \frac{1}{2} \cdot 10 = 5$

$AM = BM \cdot \sqrt{3} = 5\sqrt{3}$

The altitude also bisects $\overline{AC}$ at , so

$AC = 2 \cdot AM = 2 \cdot 5 \sqrt{3} = 10 \sqrt{3}$ .

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Example Question #2 : Dsq: Calculating The Length Of A Diagonal Of A Polygon

Calculate the diagonal of a rectangle.

Statement 1: The perimeter is 10.

Statement 2: The area is 4.

Possible Answers:

$\textup{Statement 1) ALONE is sufficient, but Statement 2) ALONE is not sufficient to answer the question.}$

$\textup{EACH statement ALONE is sufficient.}$

$\textup{Statement 2) ALONE is sufficient, but Statement 1) ALONE is not sufficient to answer the question.}$

$\textup{BOTH statements TOGETHER are NOT sufficient, and additional data is needed to answer the question. }$

$\textup{BOTH statements taken TOGETHER are sufficient to answer the question, but neither statement ALONE is sufficient.}$

Correct answer:

$\textup{BOTH statements taken TOGETHER are sufficient to answer the question, but neither statement ALONE is sufficient.}$

Explanation:

Statement 1: The perimeter is 10.

Given the perimeter of a rectangle is 10, set up the equation such that the sum of twice of length and width are equal to 10.

2L+2W=10

There are multiple combinations of length and width that will work in this scenario.

Statement 2: The area is 4.

Write the formula for area of a rectangle and substitute the area.

A=LW

4=LW

Statements 1 and 2 require each other in order to solve for the length and width since there are two unknown variables.

Afterward, the Pythagorean Theorem can be used to solve for the diagonal of the rectangle.

Therefore:

$\textup{BOTH statements taken TOGETHER are sufficient to answer the question, but neither statement ALONE is sufficient.}$

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Example Question #3 : Dsq: Calculating The Length Of A Diagonal Of A Polygon

A regular pentagon has been drawn on the side of a building by some mathematically minded graffiti artists. What is the length of a diagonal across it?

1) The length of a side is 5feet .

2) Each of the internal angles is 108 degrees.

Possible Answers:

Statement 1 alone is sufficient.

Statement 2 alone is sufficient.

Together the two statements are sufficient.

Either of the statements is sufficient.

Neither of the statements, separate or together, is sufficient.

Correct answer:

Statement 1 alone is sufficient.

Explanation:

If the pentagon is regular, then it is known that each of the internal angles is 108 degrees, since the sum of the five interior angles of a pentagon is 540 degrees. Statement 2 does not give new information, nor does it give enough.

Statement 1 does, however. The length of the diagonal can be found by using the law of cosines:

c^2=a^2+b^2-2abcos(C)

a=5, b=5, C=108

and c is the length of the diagonal.

$d=\sqrt{5^2+5^2-2(5)(5)cos(108)}$

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GMAT Math : DSQ: Calculating the length of a diagonal of a polygon

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Example Question #1 : Dsq: Calculating The Length Of A Diagonal Of A Polygon

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Example Question #3 : Dsq: Calculating The Length Of A Diagonal Of A Polygon

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