GMAT Math : Geometry

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Example Questions

Example Question #11 : Dsq: Calculating An Angle In A Polygon

A nonagon is a nine-sided polygon.

Is Nonagon ABCDEFGHI regular?

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

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

Possible Answers:

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.

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

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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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Example Question #14 : Polygons

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.

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.

BOTH statements TOGETHER are insufficient 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:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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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Example Question #267 : Data Sufficiency Questions

Pentagon

Note: Figure NOT drawn to scale.

Refer to the above figure. Is Pentagon PENTA regular?

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

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

Possible Answers:

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.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient 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:

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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Example Question #261 : Data Sufficiency Questions

Thingy_4

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate .

Statement 1: CD = 8

Statement 2: EB = 15

Possible Answers:

xxx

Either alone

Correct answer:

Either alone

Explanation:

Refer to the figure below, in which $\overline{EB}$ has been constructed, and the top and right sides of the figure have been extended to their intersection to form Rectangle AFDE .

Thingy_4

Assume Statement 1 alone. Since opposite sides of a rectangle have the same length, FD =EA= 12 . By segment addition, FC+CD= FD , and, since CD = 8 , by substitution, FC+8= 12 . Therefore, FC = 4 , and the Pythagorean Theorem can be used to find :

$BF = \sqrt{(BC)^{2} - (FC)^{2}} = \sqrt{5^{2} - 4^{2}} = \sqrt{25-16} = \sqrt{9} = 3$

Since opposite sides of a rectangle have the same length, AF = ED = 12 , and by segment addition, AB + BF = AF . By substitution, AB + 3= 12 , and AB = 9 .

Assume Statement 2 alone. Since $\overline{EB}$ , the hypotenuse of right triangle $\bigtriangleup ABE$ , and $\overline{A E}$ , one of its legs, have lengths 15 and 12, respectively, the length of the other leg $\overline{A B}$ can be found using the Pythagorean Theorem:

$AB = \sqrt{(EB)^{2} - (AE)^{2}} = \sqrt{15^{2} - 12^{2}} = \sqrt{225-144} = \sqrt{81} = 9$

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Example Question #151 : Geometry

Thingy

Note: Figure NOT drawn to scale

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

Statement 1: AB = 7

Statement 2: CD = 12

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.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER 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.

Correct answer:

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

Explanation:

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

Thingy

$\overline{BC}$ 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, EF+FD = ED = 12 . Since $\overline{AB}$ and $\overline{EF}$ are opposite sides of a rectangle, AB = EF ; similarly, GC = FD . It follows by substitution that AB+GC = 12 . Since AB = 7 , it follows that 7+GC = 12 , and GC = 5 . However, no additional information exists to find .

Assume Statement alone. By similar reasoning, BG + CD = 20 ; since CD = 12 , BG + 12= 20 , and BG = 8 . However, no information exists to find .

The two statements put together, however, yield both necessary values: BG = 8 and GC = 5 . By the Pythagorean Theorem,

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

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Example Question #152 : Geometry

Thingy

Note: Figure NOT drawn to scale.

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

Statement 1: DE = 4

Statement 2: AB = 7

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.

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.

EITHER statement ALONE is 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:

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:

Thingy

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, FG + GE = FE = 12 from segment addition, AB = 7 , from Statement 2, and from congruence of opposite sides of a rectangle, AB = FG and CD = GE . Therefore, AB + CD = 12 , 7 + CD = 12 , and CD = 5 .

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Example Question #4 : Dsq: Calculating The Length Of A Side Of A Polygon

Given a regular hexagon HEXAGN , 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.

Possible Answers:

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.

BOTH statements TOGETHER are insufficient 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:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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

Hexagon_2

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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Example Question #5 : Dsq: Calculating The Length Of A Side Of A Polygon

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

Statement 1: EX= 7 .

Statement 2: Hexagon HEXAGO has perimeter 42.

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 insufficient to answer the question.

Explanation:

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, EX= 7 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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Example Question #6 : Dsq: Calculating The Length Of A Side Of A Polygon

Give the length of side $\overline{PE}$ of Pentagon PENTA .

Statement 1: $\overline{PE} \cong\overline{PA} \cong \overline{NT}$

Statement 2: $\overline{NE}$ and $\overline{TN}$ both have length 10.

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.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 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.

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 states that $\overline{PE}$ is congruent to two other sides, but gives no actual measurements. Statement 2 alone gives the actual measurements of two other segments, but without further information, such as how their lengths relates to that of $\overline{PE}$ , no information about $\overline{PE}$ can be inferred.

Now, assume both statements to be true. From Statement 2, $\overline{TN}$ has length 10, and from Statement 2, $\overline{NT}$ , which is the same line segment (which can be named after its endpoints in either order), has the same length as $\overline{PE}$ . Therefore, PE = 10 .

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Example Question #7 : Dsq: Calculating The Length Of A Side Of A Polygon

Give the length of side $\overline{PE}$ of Pentagon PENTA .

Statement 1: Pentagon PENTA has perimeter 50.

Statement 2: EP = 10

Possible Answers:

BOTH statements TOGETHER give sufficient information to answer the question, but neither statement ALONE gives sufficient information to answer the question.

BOTH statements TOGETHER do not give sufficient information to answer the question.

Statement 2 ALONE gives sufficient information to answer the question, but Statement 2 ALONE does NOT sufficient information to answer the question.

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

EITHER statement ALONE gives sufficient information to answer the question.

Correct answer:

Statement 2 ALONE gives sufficient information to answer the question, but Statement 2 ALONE does NOT sufficient information to answer the question.

Explanation:

Statement 1 alone only gives the perimeter - the sum of the lengths of the sides - but gives no information about the individual sidelengths. (In particular, there is no indication that the pentagon is regular).

Assume Statement 2 alone. $\overline{PE}$ and $\overline{EP}$ are two names for the same line segment, which can be named after its endpoints in either order. Therefore, PE = EP = 10 .

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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #11 : Dsq: Calculating An Angle In A Polygon

Example Question #14 : Polygons

Example Question #267 : Data Sufficiency Questions

Example Question #261 : Data Sufficiency Questions

Example Question #151 : Geometry

Example Question #152 : Geometry

Example Question #4 : Dsq: Calculating The Length Of A Side Of A Polygon

Example Question #5 : Dsq: Calculating The Length Of A Side Of A Polygon

Example Question #6 : Dsq: Calculating The Length Of A Side Of A Polygon

Example Question #7 : Dsq: Calculating The Length Of A Side Of A Polygon

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