Polygons - GMAT Quantitative

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Question

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 $\overline{AC}$ ?

Answer

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

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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Question

Calculate the diagonal of a rectangle.

Statement 1: The perimeter is 10.

Statement 2: The area is 4.

Answer

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.

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.

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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Question

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?

The length of a side is .

Each of the internal angles is degrees.

Answer

If the pentagon is regular, then it is known that each of the internal angles is degrees, since the sum of the five interior angles of a pentagon is 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:

and c is the length of the diagonal.

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

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Question

How many sides does a regular polygon have?

Each of the angles measures 140 degrees.

Each of the sides has measure 8.

Answer

The relationship between the number of sides of a regular polygon and the measure of a single angle is

$A = \frac{180(n-2)}{n}$

If we are given that , then we can substitute and solve for :

$140 = \frac{180(n-2)}{n}$

$140 = \frac{180n-360}{n}$

making the figure a nine-sided polygon.

Knowing only the measure of each side is neither necessary nor helpful; for example, it is possible to construct an equilateral triangle with sidelength 8 or a square with sidelength 8.

The correct answer is that Statement 1 alone is sufficient to answer the question, but not Statement 2 alone.

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Question

Note: Figure NOT drawn to scale.

Is the pentagon in the diagram above a regular pentagon?

Statement 1: $m \angle 1= 70^{\circ }$

Statement 2: The hexagon in the diagram is not a regular hexagon.

Answer

Information about the hexagon is irrelevant, so Statement 2 has no bearing on the answer to the question.

The measures of the exterior angles of any pentagon, one per vertex, total $360^{\circ }$ , and they are congruent if the pentagon is regular, so if this is the case, each would measure $360 \div 5 = 72^{\circ }$ . But if Statement 1 is true, then an exterior angle of the pentagon measures $70^{\circ}$ . Therefore, Statement 1 is enough to answer the question in the negative.

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Question

Note: Figure NOT drawn to scale.

$m \angle A = m \angle C = m \angle E = 119$

What is $m \angle D$ ?

Statement 1: $m \angle B$ , $m \angle D$ , and $m \angle F$ are the first three terms, in order, of an arithmetic sequence.

Statement 2: $m \angle B + 2 = m \angle F$

Answer

The sum of the measures of the angles of a hexagon is $180 (6-2) = 720^{\circ }$

Therefore,

$m \angle A + m \angle B + m \angle C + m \angle D + m \angle E + m \angle F = 720$

$\ 119 + m \angle B + 119 + m \angle D + 119 + m \angle F = 720$

$m \angle B + m \angle D + m \angle F + 357 = 720$

$m \angle B + m \angle D + m \angle F = 720 - 357 = 363$

Suppose we only know that $m \angle B$ , $m \angle D$ , and $m \angle F$ are the first three terms of an arithmetic sequence, in order. Then for some common difference ,

$m \angle B = m \angle D - d$

$\ m \angle F = m \angle D + d$

$\ \left (m \angle D - d \right )+ m \angle D \left + ( m \angle D +d \right ) = 363$

$\ 3 \cdot m \angle D \left = 363$

$m \angle D \left = 121$

Suppose we only know that $m \angle B + 2 = m \angle F$ . Then

$m \angle B + m \angle D + \left (m \angle B + 2 \right ) = 363$

$2 \cdot m \angle B + m \angle D + 2 = 363$

$2 \cdot m \angle B + m \angle D = 361$

With no further information, we cannot determine $m \angle D$ .

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Question

What is the sum of six numbers, ?

Statement 1: are the measures of the interior angles of a hexagon.

Statement 2: The median of the data set $\left {}a,b,c,d,e,f \right }$ is 120.

Answer

The sum of the measures of the six interior angles of any hexagon is , so if Statement 1 is true, then the sum of the six numbers is 720, regardless of the individual values.

Statement 2 is not enough, however, for us to deduce this sum; this only tells us the mean of the third- and fourth-highest values.

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Question

What is the sum of five numbers, ?

Statement 1: are the measures of the interior angles of a pentagon.

Statement 2: The mean of the data set $\left {}a,b,c,d,e\right }$ is 108.

Answer

The sum of the measures of the five interior angles of any pentagon is , so if Statement 1 is true, then the sum of the five numbers is 540, regardless of the individual values.

The mean of five data values multiplied by 5 is the sum of the values, so if Statement 2 is true, then their sum is $108 \cdot 5 = 540$ .

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Question

Note: Figure NOT drawn to scale.

The diagram above shows a triangle and a rhombus sharing a side. Is that rhombus a square?

Statement 1: The triangle is not equilateral.

Statement 2: $m \angle AVC = 150^{\circ }$

Answer

To show that the rhombus is a square, you need to demonstrate that one of its angles is a right angle - that is, $90 ^{\circ }$ . Both statements together are insufficent - if $m \angle AVC = 150^{\circ }$ , you would need to demonstrate that $m \angle AVD = 60^{\circ }$ is true or false, and the fact that the triangle is not equilateral is not enough to prove or to disprove this.

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Question

What is the measure of an interior angle of a regular polygon?

Statement 1: The polygon has 20 sides.

Statement 2: An exterior angle of the polygon measures $18^{\circ}$ .

Answer

From Statement 1, you can calculate the measure of an interior angle as follows:

$\frac{180(20-2) }{20} = \frac{180(18) }{20} = 162 ^{\circ }$

From Statement 2, since an interior angle and an exterior angle at the same vertex form a linear pair, they are supplementary, so you can subtract 18 from 180:

$180 - 18 = 162^{\circ }$

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Question

Is $\textrm{ polygon } ABCDE$ a regular pentagon?

Statement 1: $m \angle A > m \angle B$

Statement 2:

Answer

All of the interior angles of a regular polygon are congruent, as are all of its sides. Statement 1 violates the former condition; statement 2 violates the latter.

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Question

Is the degree measure of an exterior angle of a regular polygon an integer?

Statement 1: The number of sides of the polygon is divisible by 7.

Statement 2: The number of sides of the polygon is divisible by 10.

Answer

The sum of the degree measures of the exterior angles, one per vertex, of any polygon is 360, so each exterior angle of a regular polygon with sides measures $\frac{360}{N}$ .

For $\frac{360}{N}$ to be an integer, every factor of must be a factor of 360. This does not happen if 7 is a factor of , so Statement 1 disproves this. This may or may not happen if 10 is a factor of - $\frac{360}{40} = 9$ , but $\frac{360}{80} = 4.5$ , so Statement 2 does not provide an answer.

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Question

Is the degree measure of an exterior angle of a regular polygon an integer?

Statement 1: The number of sides of the polygon is a factor of 30.

Statement 2: The number of sides of the polygon is a factor of 40.

Answer

The sum of the degree measures of the exterior angles, one per vertex, of any polygon is 360, so each exterior angle of a regular polygon with sides measures $\frac{360}{N}$ . Therefore, the measure of one exterior angle of a regular polygon is an integer if and only if is a factor of 360.

If is a factor of a factor of 360, however, then is a factor of 360. 30 and 40 are both factors of 360: $360 \div 30 = 12$ and $360 \div 40 = 9$ . Therefore, it follows from either statement that the number of sides is a factor of 360, and each exterior angle has a degree measure that is an integer.

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Question

What is the measure of $\angle 1$ ?

Statement 1: $\angle 1$ is an exterior angle of an equilateral triangle.

Statement 2: $\angle 1$ is an interior angle of a regular hexagon.

Answer

An exterior angle of an equilateral triangle measures $360 ^{\circ } \div 3 = 120 ^{\circ }$ . An interior angle of a regular hexagon measures $\left [180 (6-2) \div 6 \right ]^ {\circ} = 120 ^ {\circ}$ . Either statement is sufficient.

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Question

A nonagon is a nine-sided polygon.

Is Nonagon regular?

Statement 1: $m \angle A = 150 ^{\circ }$

Statement 2: $m \angle G = 150 ^{\circ }$

Answer

Each angle of a regular nonagon measures $\left [\frac{180 (9-2) }{9 } \right ]^ {\circ } = 140^ {\circ }$

Therefore, each of the two statements proves that the nonagon is not regular.

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Question

Note: Figure NOT drawn to scale.

Refer to the above figure. Is Pentagon regular?

Statement 1: $m \angle ENY = 72 ^{\circ }$

Statement 2: $m \angle TAX = 70 ^{\circ }$

Answer

If a shape is regular, that means that all of its sides are equal. It also means that all of its interior angles are equal. Finally, if all of the interior angles are equal, then the exterior angles will all be equal to each other as well.

In a regular polygon, we can find the measure of ANY exterior angle by using the formula

$\frac{360}{n}$

where is equal to the number of sides.

Each exterior angle of a regular (five-sided) pentagon measures

$\left (\frac{360 }{5 } \right)^ {\circ } = 72^ {\circ }$

Statement 1 alone neither proves nor disproves that the pentagon is regular. We now know that one exterior angle is $72^{\circ}$ , but we do not know if any of the other exterior angles are also $72^ {\circ }$ .

Statement 2, however, proves that the pentagon is not regular, as it has at least one exterior angle that does not have measure $72^ {\circ }$ .

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Question

Note: Figure NOT drawn to scale.

Refer to the above figure. What is the length of $\overline{CD}$ ?

Statement 1:

Statement 2:

Answer

To find the length of $\overline{CD}$ , we can extend $\overline{BC}$ to meet $\overline{EF}$ at a point to form two rectangles, as seen below:

Statement 1 gives no helpful information, since the length of $\overline{DE}$ , which is not parallel to $\overline{CD}$ , has no bearing on that side's length.

If we are given Statement 2 alone, then, as seen in the diagram, from segment addition, , from Statement 2, and from congruence of opposite sides of a rectangle, and . Therefore, , , and .

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Question

Given a regular hexagon , what is the length of $\overline{HE}$ ?

Statement 1: The hexagon is circumscribed by a circle with circumference $12 \pi$ .

Statement 2: $\overline{HA }$ hs length 12.

Answer

Below is a regular hexagon , with its three diameters, its center , and its circumscribed circle, which also has center .

If Statement 1 is true. then the circle, with circumference $12 \pi$ , has as its diameter $12 \pi \div \pi = 12$ , which is 12; this makes the two statements equivalent, so we need only establish that one statement is sufficient or insufficient.

Either way, , the radius of the hexagon, is 6. The six triangles that are formed by the sides and diameters of a regular hexagon are all equilateral by symmetry, so each side of the hexagon - in particular, $\overline{HE}$ - has length 6.

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Question

Note: Figure NOT drawn to scale

Refer to the above figure. Give the length of $\overline{BC}$ .

Statement 1:

Statement 2:

Answer

We can construct perpendicular line segments from to $\overline{ED}$ and from to $\overline{AE}$ as follows:

$\overline{BC}$ is the hypotenuse of a right triangle $\bigtriangleup BGC$ , so if we can determine the lengths of $\overline{BG}$ and $\overline{GC}$ , we can use the Pythagorean Theorem to determine the length of $\overline{BC}$ .

Assume Statement 1 alone. By segment addition, . Since $\overline{AB}$ and $\overline{EF}$ are opposite sides of a rectangle, ; similarly, . It follows by substitution that . Since , it follows that , and . However, no additional information exists to find .

Assume Statement alone. By similar reasoning, ; since , , and . However, no information exists to find .

The two statements put together, however, yield both necessary values: and . By the Pythagorean Theorem,

$BC = \sqrt{(BG)^{2}+(GC)^{2}} = \sqrt{8^{2}+5^{2}} = \sqrt{64+25} = \sqrt{89}$ .

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Question

Give the length of side $\overline{HE}$ of Hexagon .

Statement 1: .

Statement 2: Hexagon has perimeter 42.

Answer

Assume both statements to be true, and examine these two scenarios:

Case 1: The hexagon could have six sides of length 7.

Case 2: The hexagon has four sides of length 7, one of which is $\overline{EX}$ , one side of length 6, and one side - $\overline{HE}$ - of length 8.

In both situations, and the perimeter of the hexagon is 42:

$7 \times 6= 7 \times 4+6+8 = 42$ .

The conditions of both statements would be met in both scenarios, so the two statements together are insufficient.

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