Geometry - GMAT Quantitative

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Question

Let be a positive integer.

Evaluate $i^{N}$ .

Statement 1: is a multiple of 3.

Statement 2: is a multiple of 7.

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Answer

Assume that both statements are true. The value of $i^{N}$ , a positive integer, is equal to $i^{R}$ , where is the remainder of the division of by 4, so we can use this fact to show that insufficient information is provided.

Case 1: .

$21 \div 4 = 5\textup{ R }1$ , so

$i^{N} = i^{21} = i^{1}= i$

Case 2: .

$42 \div 4 = 10 \textup{ R }2$ , so

$i^{N} = i^{42} = i^{2}= -1$

In both cases, both statements are true, but the value of $i^{N}$ differs.

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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.

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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.

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

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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

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

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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 $18^{\circ}$ .

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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:

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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.

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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.

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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.

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

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

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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

What is the area of a parallelogram with four equal sides?

(1) Each side is $\dpi{100} \small 5cm$

(2) One diagonal is $\dpi{100} \small 6cm$

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Answer

$\dpi{100} \small S= \frac{1}{2}\times L1\times L2$ ( $\dpi{100} \small L1$ and $\dpi{100} \small L2$ are the diagonals of the rhombus). From statement (1) we cannot get to the lengths of diagonals. From statement (2) we only know the length of one diagonal, which is insufficient. However, putting the two statements together, we can use the Pythagorean Theorem to calculate the other diagonal, and then use the formula to calculate the area.

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Question

What is the area of a regular hexagon?

Statement 1: The perimeter of the hexagon is 48.

Statement 2: The radius of the hexagon is 8.

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Answer

A regular hexagon can be viewed as the composite of six equilateral triangles. The sidelength of each of the triangles is equal to both the sidelength of the hexagon (one-sixth of the perimeter) and the radius of the hexagon. From either statement, it is possible to find the area of one triangle by substituting in the area formula

$A = \frac{s^{2}\sqrt{3}}{4}$

and multiplying the result by 6.

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Question

What is the area of a trapezoid?

Statement 1: The length of its midsegment is 20.

Statement 2: The lengths of its bases are 18 and 22.

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Answer

The area of a trapezoid can be found using the following formula:

$A = \frac{1}{2} (B + b) h$

Where are the lengths of the bases and is the height. From either statement, can be determined, but is not given in either statement.

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Question

What is the area of a regular hexagon?

Statement 1: The area of the circle inscribed inside the hexagon is $108 \pi$ .

Statement 2: The circumference of the circle that is circumscribed about the hexagon is $24 \pi$ .

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Answer

A regular hexagon can be viewed as a composite of six equilateral triangles, each of whose sidelength is the radius - the distance from its center to a vertex - of the hexagon. If the radius of the hexagon is known, then the area of the hexagon can be calculated to be $A = 6 \cdot \frac{r^{2}\sqrt{3}}{4} = \frac{3r^{2}\sqrt{3}}{2}$ .

From Statement 1 alone, the radius of the inscribed circle, or incircle, can be calculated from the area formula (by dividing the area by $\pi$ and extracting the square root). This length coincides with the height of each equilateral triangle. From there, the 30-60-90 Theorem can be used to find the sidelength of each triangle, and the area of the hexagon follows.

From Statement 2 alone, the radius of the circumscribed circle, or circumcircle, can be found by dividing its circumference by $2 \pi$ . This radius coincides with the radius of the hexagon, and the area can be calculated from there.

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Question

Give the area of the above regular octagon.

Statement 1: The circle that circumscribes Quadrilateral has area $32 \pi$ .

Statement 2: $\Delta BXC$ has area 16.

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Answer

Quadrilateral is a square; each of its diagonals is a diameter of its circumscribed circle, or circumcircle. Therefore, if we know the area of its circumcircle from Statement 1 to be $32 \pi$ , we can calculate the radius from the area formula (divide by $\pi$ , extract the square root of the quotient). Twice this is the diameter, which is also the length of a diagonal of this square. Divide this by $\sqrt{2}$ to get . This is also equal to , the length of one side; this is sufficient to get the area of the octagon.

From Statement 2 alone, since the area of isosceles triangle $\Delta BXC$ is known to be 8, the length of each leg can be found using the formula

$\frac{1}{2} s^{2} = 16$

Since $\Delta BXC$ is a 45-45-90 triangle, multiply this leg length by $\sqrt{2}$ to get , the length of one side; this is sufficient to get the area of the octagon.

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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 $\overline{AC}$ ?

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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.

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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.

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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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