GMAT Math : DSQ: Understanding rays

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Example Questions

Example Question #6 : Geometry

Lines

Note: Figure NOT drawn to scale.

Evaluate .

Statement 1: AD = 24

Statement 2: BE = 24

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.

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.

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 #2 : 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:

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

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient 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 #1 : Dsq: Understanding Rays

, , 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:

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 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.

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 #8 : 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 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.

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:

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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Example Question #1 : Dsq: Understanding Rays

, , and are distinct points.

True or false: $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray.

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

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

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 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:

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:

Rays

In both figures, $AB = 2 \cdot AC$ , but only in the first figure, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray.

Assume Statement 2 alone. If is the midpoint of $\overline{AB}$ , must be on $\overrightarrow{AB}$ , as in the top figure, so $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are one and the same.

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Example Question #2 : Dsq: Understanding Rays

, , and are distinct points.

True or false: $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray.

Statement 1: AB+BC > AC

Statement 2: AB+ AC > BC .

Possible Answers:

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.

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:

We show that both statements together provide insufficient information by giving two scenarios in which both statements are true.

Case 1: , , and are noncollinear. The three points are vertices of a triangle, and by the Triangle Inequality Theorem,

AB+BC > AC and

AB + AC > BC .

Also, since the three points are not on a single line, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are parts of different lines and cannot be the same ray.

Case 2: $\overline{AB}$ with length 2 and midpoint .

Rays

AB+BC =2+1 = 3 and AC = 1 , so AB+BC > AC ; similarly, AB + AC > BC . Also, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray, since they have the same endpoint and is on $\overrightarrow{AB}$ .

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Example Question #3 : Dsq: Understanding Rays

, , and are distinct points.

True or false: $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray.

Statement 1: , , and are collinear.

Statement 2: AB + BC = AC .

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.

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

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 prove the rays to be the same or different, as seen in these diagrams:

Rays

In both figures, , , and are collinear, satisfying the condition of Statement 1. But In the top figure, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray, since is on $\overrightarrow{AB}$ ; in the bottom figure, since is not on $\overrightarrow{AB}$ , $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are distinct rays.

Assume Statement 2 alone. Suppose $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are not the same ray. Then one of two things happens:

Case 1: , , and are noncollinear. The three points are vertices of a triangle, and by the triangle inequality,

AB + BC > AC ,

contradicting Statement 2.

Case 2: , , and are collinear. must be between and , as in the bottom figure, since if it were not, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ would be the same ray. By segment addition,

AB + AC = BC

AB + AC+AB = BC+AB

$AB + BC = AC + 2 \cdot AB > AC$ ,

contradicting Statement 2.

By contradiction, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray.

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GMAT Math : DSQ: Understanding rays

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #6 : Geometry

Example Question #2 : Geometry

Example Question #1 : Dsq: Understanding Rays

Example Question #8 : Geometry

Example Question #1 : Dsq: Understanding Rays

Example Question #2 : Dsq: Understanding Rays

Example Question #3 : Dsq: Understanding Rays

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