GMAT Math : Geometry

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Example Question #1 : Lines

Data Sufficiency Question

Is Line A perpendicular to the following line?

$y=-\frac{1}{2}x+3$

Statement 1: The slope of Line A is 3.

Statement 2: Line A passes through the point (2,3).

Possible Answers:

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question.

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

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

Each statement alone is sufficient.

Correct answer:

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

Explanation:

To determine if two lines are perpendicular, only the slope needs to be considered. The slopes of perpendicular lines are the negative reciprocals of each other. Knowing a single point on the line is not sufficient, as an infinite number of lines can pass through and individual point.

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Example Question #2 : Geometry

Untitled

Refer to the above figure. True or false: $m \perp n$

Statement 1: $\triangle AXB$ is equilateral.

Statement 2: Line bisects $\angle AXB$ .

Possible Answers:

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.

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.

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 establishes nothing about the angle makes with , as it is not part of the triangle. Statement 2 alone only establishes that AY = YB .

Assume both statements are true. Then $\overleftrightarrow{XY}$ is an altitude of an equilateral triangle, making it - and - perpendicular with the base $\overline {AB}$ - and .

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Example Question #1 : Geometry

Untitled Statement 1:

Refer to the above figure. Are the lines perpendicular?

Statement 1: x = 9

Statement 2: x= y+1

Possible Answers:

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 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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. The measure of one of the angles formed is

$x^{2}+ 9 = 9^{2}+ 9= 81 + 9 = 90$ degrees.

Assume Statement 2 alone.

By substituting x-1 for , one angle measure becomes

11y + 2 = 11 (x-1) +2= 11x-11+2= 11x-9

The marked angles are a linear pair and thus their angle measures add up to 180 degrees; therefore, we can set up an equation:

$\left ( 11x-9 \right )+ \left ( x^{2}+9 \right ) = 180$

$x^{2}+11x = 180$

$x^{2}+11x - 180 = 0$

(x+20)(x-9) = 0

x= -20 or x= 9

x= -20 yields illegal angle measures - for example,

$x^{2}+ 9 = (-20)^{2}+ 9= 409$

x= 9 yields angle measures $90 ^{\circ }$ for both angles; the angles are right and the lines are perpendicular.

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Example Question #1 : Geometry

Transversal

Refer to the above figure.

True or false: $m \perp t$

Statement 1: $m \angle 2 = 89^{\circ }$

Statement 2: $\angle 3 \cong \angle 6$

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 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 establishes by definition that $l \perp \! \! \! \! \! / \; t$ , but does not establish any relationship between and .

By Statement 2 alone, since alternating interior angles are congruent, l || m , but no conclusion can be drawn about the relationship of , since the actual measures of the angles are not given.

Assume both statements are true. By Statement 2, l || m . $\angle 2$ and $\angle 6$ are corresponding angles formed by a transversal across parallel lines, so $m \angle 6 = m \angle 2 = 89^{\circ }$ . $\angle 6$ is not a right angle, so $m\perp \! \! \! \! \! / \; t$ .

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Example Question #5 : Geometry

Untitled

Refer to the above figure. True or false: $m \perp n$

Statement 1: $\bigtriangleup XAY \cong \bigtriangleup XBY$

Statement 2: Line bisects $\angle AXB$ .

Possible Answers:

EITHER statement ALONE is 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.

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.

Correct answer:

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

Explanation:

Assume Statement 1 alone. Then, as a consequence of congruence, $\angle XYA$ and $\angle XYB$ are congruent. They form a linear pair of angles, so they are also supplementary. Two angles that are both congruent and supplementary must be right angles, so $m \perp n$ .

Assume Statement 2 alone. Then $\angle AXY \cong \angle BXY$ , but without any other information about the angles that $\overleftrightarrow{AX}, \overleftrightarrow{BX},$ or make with , it cannot be determined whether $m \perp n$ or not.

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Example Question #6 : Geometry

The equations of two lines are:

4x+5y = 20

Ax+8y=B

Are these lines perpendicular?

Statement 1: A=4

Statement 2: B = -20

Possible Answers:

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 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.

Correct answer:

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

Explanation:

The lines of the two equations must have slopes that are the opposites of each others reciprocals.

Write each equation in slope-intercept form:

4x+5y = 20

5y = -4x+20

$y = -\frac{4}{5} x+4$

$m_{1}= -\frac{4}{5}$

Ax+8y=B

8y=-Ax+B

$y=-\frac{A}{8}x+\frac{B}{8}$

$m_{2}= -\frac{A}{8}$

As can be seen, knowing the value of is necessary and sufficient to answer the question. The value of is irrelevant.

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

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Example Question #7 : Geometry

Lines

Note: Figure NOT drawn to scale.

Evaluate .

Statement 1: AD = 24

Statement 2: BE = 24

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:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Even with both statements, cannot be determined because the length of is missing.

For example, we can have AB = DE = 8 and BD=16 , making AE = 32 ; or, we can have AB = DE = 10 and BD=14 , making AE = 34 . Neither scenario violates the conditions given.

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Example Question #8 : Geometry

, , and are distinct points.

True or false: $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are opposite rays.

Statement 1: is on $\overleftrightarrow{AC}$

Statement 2: is on $\overleftrightarrow{AB}$

Possible Answers:

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.

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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Both statements are equivalent, as both are equivalent to stating that , , and are collinear. Therefore, it suffices to determine whether the fact that the points are collinear is sufficient to answer the question.

Rays

In both of the above figures, , , and are collinear, so the conditions of both statements are met. But in the top figure, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray, since is on $\overrightarrow{AB}$ ; in the bottom figure, since and are on opposite sides of , $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are opposite rays.

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Example Question #9 : Geometry

, , and are distinct points.

True or false: $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are opposite rays.

Statement 1: $AC = 2 \cdot AB$ .

Statement 2: is the midpoint of $\overline{AC}$ .

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

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 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:

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

Explanation:

We show Statement 1 alone is insufficient to determine whether the two rays are the same by looking at the figures below. In the first figure, is the midpoint of $\overline{AC}$ .

Rays

In both figures, $AC = 2 \cdot AB$ . But only in the second figure, and are on the opposite side of the line from , so only in the second figure, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are opposite rays.

Assume Statement 2 alone. If is the midpoint of $\overline{AC}$ , then, as seen in the top figure, is on $\overrightarrow{AC}$ . Therefore, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray, not opposite rays.

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Example Question #1 : Geometry

, , and are distinct points.

True or false: $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are opposite rays.

Statement 1: AB+BC > AC

Statement 2: AB+ AC > BC

Possible Answers:

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.

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.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER 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:

Statement 1 alone does not answer the question.

Case 1: Examine the figure below.

Rays

AB+BC= AB + AB + AC > AC ,

thereby meeting the condition of Statement 1.

Also, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are opposite rays, since and are on opposite sides of the same line from .

Case 2: Suppose , , and are noncollinear.

The three points are vertices of a triangle, and by the Triangle Inequality Theorem,

AB+BC > AC .

Furthermore, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are not part of the same line and are not opposite rays.

Now assume Statement 2 alone. As can be seen in the diagram above, if $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are opposite rays, then by segment addition, AB+BC = AC , making Statement 2 false. Contrapositively, if Statement 2 holds, and $AB+ AC \ne BC$ , then $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are not opposite rays.

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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Lines

Example Question #2 : Geometry

Example Question #1 : Geometry

Example Question #1 : Geometry

Example Question #5 : Geometry

Example Question #6 : Geometry

Example Question #7 : Geometry

Example Question #8 : Geometry

Example Question #9 : Geometry

Example Question #1 : Geometry

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