GMAT Math : Geometry

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Example Questions

Example Question #8 : Dsq: Calculating The Length Of A Side Of A Polygon

True or false: $\overline{NT}$ is the shortest side of Pentagon PENTA .

Statement 1: Pentagon PENTA has perimeter 65.

Statement 2: NT = 12 .

Possible Answers:

Statement 1 ALONE provides sufficient information to answer the question, but Statement 2 ALONE does NOT provide sufficient information to answer the question.

BOTH statements TOGETHER provide insufficient information to answer the question.

Statement 2 ALONE provides sufficient information to answer the question, but Statement 1 ALONE does NOT provide sufficient information to answer the question.

EITHER Statement ALONE provides sufficient information to answer the question.

BOTH statements TOGETHER provide sufficient information to answer the question, but NEITHER statement ALONE provides sufficient information to answer the question.

Correct answer:

BOTH statements TOGETHER provide insufficient information to answer the question.

Explanation:

Assume both statements are true. If the pentagon has sides of length 12, 13, 13, 13, and 14, with $\overline{NT}$ the side of length 12, then $\overline{NT}$ is the shortest side, and the perimeter is

12+13+13+13+14 = 65 .

On the other hand, if the pentagon has sides of length 11, 12, 13, 14, and 15,

with $\overline{NT}$ the side of length 12, then $\overline{NT}$ is not the shortest side, and the perimeter is

11+ 12+13+14 +15= 65 .

The two statements together are insufficient.

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Example Question #9 : Dsq: Calculating The Length Of A Side Of A Polygon

True or false: $\overline{XA}$ is the longest side of Hexagon HEXAGO .

Statement 1: XA = 10

Statement 2: Hexagon HEXAGO has perimeter 66.

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

Either statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Statement 1 alone only gives information about one side of the hexagon, and Statement 2 gives only information about the perimeter without giving any clues as to the individual sidelengths; neither is sufficient to answer the question.

Assume both statements are true. If $\overline{XA}$ , with length 10, is the longest side of Hexagon HEXAGO , then

HE, EX, AG, GO, OH < 10

By the addition property of inequality,

HE+ EX+XA+ AG+ GO+ OH < 10 +10 +10 +10 +10 +10 = 60 < 66

This means the sum of the sidelengths of the hexagon, which is its perimeter, is less than 66, in contradiction to Statement 2. $\overline{XA}$ cannot be the longest side.

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Example Question #31 : Polygons

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

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

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

Possible Answers:

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Correct answer:

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Explanation:

$\dpi{100} \small S= \frac{1}{2}\times L1\times L2$ ( $\dpi{100} \small L1$ 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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Example Question #2 : Dsq: Calculating The Area Of A Polygon

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.

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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 s = 8 in the area formula

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

and multiplying the result by 6.

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Example Question #3 : Dsq: Calculating The Area Of A Polygon

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.

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

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

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

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

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Example Question #32 : Polygons

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 $24 \pi$ .

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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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Example Question #282 : Data Sufficiency Questions

Octagon_1

Give the area of the above regular octagon.

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

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

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Quadrilateral WXYZ 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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Example Question #161 : Geometry

Find the length of the edge of a tetrahedron.

Statement 1: The volume is 6.

Statement 2: The surface area is 6.

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Statement 1:) The volume is 6.

Write the formula to find the edge of the tetrahedron given the volume.

$V=\frac{e^3}{6\sqrt2}$

Given the volume, it is possible to find the edge of the tetrahedron.

Statement 2:) The surface area is 6.

Write the formula to find the edge of the tetrahedron given the surface.

$SA=\sqrt3 e^2$

Substitute the surface area to find the edge.

Therefore:

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

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Example Question #1 : Tetrahedrons

Pyramid 1 in three-dimensional coordinate space has as its base the square with vertices at the origin, (10,0,0) , (10,10,0) , and (0,10,0) , and its apex at the point (D,E,F) ; Pyramid 2 has as its base the square with vertices at the origin, (12,0,0) , (12,12,0) , and (0,12,0) , and its apex at the point (G,H,J) . All six variables represent positive quantities. Which pyramid has the greater volume?

Statement 1: $10 \le F \le 12$ and $10 \le J \le 12$

Statement 2: F< J

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

The volume of a pyramid is one third the product of its height and the area its base. The two pyramids have the same base, so the pyramid with the greater height will have the greater volume (and if their heights are equal, their volumes are equal).

Pyramid 1 is shown below:

Pyramid

The base of the pyramid is on the -plane, so the height of the pyramid is the perpendicular distance from apex (D,E,F) to this plane; this is the -coordinate, . The base of the pyramid is a square of sidelength 10, so its area is the square of 10, or 100. This makes the volume of Pyramid 1

$V_{1} = \frac{1}{3} \cdot 100 \cdot F = \frac{100}{3}F = \left ( 33 \frac{1}{3}\right )F$

Similarly, the volume of Pyramid 2 is

$V_{2} = \frac{1}{3} \cdot 12 ^{2}\cdot J= \frac{1}{3} \cdot 144 \cdot J = 36 J$

Therefore, the problem asks us to determine which of $\left ( 33 \frac{1}{3}\right )F$ and 36J is the greater.

Assume Statement 1 alone. Since $10 \le F \le 12$ , we can multiply all expressions by $33 \frac{1}{3}$ to get a range for the volume of Pyramid 1:

$10 \le F \le 12$

$33 \frac{1}{3} \cdot 10 \le 33 \frac{1}{3} \cdot F \le 33 \frac{1}{3} \cdot 12$

$333 \frac{1}{3} \le V_{1} \le 400$

Similarly, since $10 \le J \le 12$ , we can multiply all expressions by 36 to get a range of values for the volume of Pyramid 2:

$10 \le J \le 12$

$36 \cdot 10 \le 36 \cdot J \le 36 \cdot 12$

$360 \le V_{2} \le 432$

Since the two ranges share values, it cannot be determined for certain which pyramid has the greater volume.

Assume Statement 2 alone. Then, since $33 \frac{1}{3} < 36$ and F < J , it easily follows that

$33 \frac{1}{3} \cdot F < 36 \cdot J$ ,

and, subsequently, Pyramid 2 has the greater volume.

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Example Question #2 : Dsq: Calculating The Volume Of A Tetrahedron

Pyramid_2

Note: Figure NOT drawn to scale, but you may assume AE < EF and EH < EF .

In the above figure, a pyramid with a rectangular base is inscribed inside a rectangular prism; its vertices are A,E,F,G,H . What is the volume of the pyramid?

Statement 1: 30-60-90 triangle $\bigtriangleup AEF$ has area $50 \sqrt{3}$ .

Statement 2: Isosceles right triangle $\bigtriangleup AEH$ has area 50.

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

The volume of the pyramid is one third the product of height and the area of its rectangular base, which is $EF \cdot EH$ ; that is,

$V = \frac{1}{3} \cdot AE \cdot EF \cdot EH$

Assume Statement 1 alone. $\bigtriangleup AEF$ has area $50 \sqrt{3}$ , which is half the product of the length of shorter leg $\overline{AE}$ and longer leg $\overline{EF}$ . Also, by the 30-60-90 Theorem, $EF = AE \cdot \sqrt{3}$ , so, combining these statements,

$\frac{1}{2} \cdot AE \cdot EF = 50 \sqrt{3}$

$\frac{1}{2} \cdot AE \cdot AE \cdot \sqrt{3} = 50 \sqrt{3}$

$\frac{\sqrt{3}}{2} \cdot (AE) ^{2} = 50 \sqrt{3}$

$\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{2} \cdot (AE) ^{2} = \frac{2}{\sqrt{3}} \cdot 50 \sqrt{3}$

$(AE) ^{2} =100$

AE = 10 , and $EF = 10 \sqrt{3}$ .

However, we do not have any way of finding out , so the volume cannot be calculated.

Assume Statement 2 alone. $\bigtriangleup AEH$ is isosceles, so AE = EH ; again, since the area of a right triangle is half the product of the lengths of its legs,

$\frac{1}{2} \cdot AE \cdot EH = 50$

$\frac{1}{2} \cdot AE \cdot AE = 50$

$\frac{1}{2} \cdot (AE )^{2}= 50$

$\frac{1}{2} \cdot (AE )^{2} \cdot 2 = 50 \cdot 2$

$(AE )^{2} =100$

AE = 10

EH = 10

However, we have no way of finding out .

The two statements together give all three of , , and , so the volume can be calculated as

$V = \frac{1}{3} \cdot 10 \cdot 10 \sqrt{3}\cdot 10 = \frac{1,000 \sqrt{3}}{3}$

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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #8 : Dsq: Calculating The Length Of A Side Of A Polygon

Example Question #9 : Dsq: Calculating The Length Of A Side Of A Polygon

Example Question #31 : Polygons

Example Question #2 : Dsq: Calculating The Area Of A Polygon

Example Question #3 : Dsq: Calculating The Area Of A Polygon

Example Question #32 : Polygons

Example Question #282 : Data Sufficiency Questions

Example Question #161 : Geometry

Example Question #1 : Tetrahedrons

Example Question #2 : Dsq: Calculating The Volume Of A Tetrahedron

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