# GMAT Math : DSQ: Understanding rays

## Example Questions

### Example Question #1 : Dsq: Understanding Rays

Note: Figure NOT drawn to scale.

Evaluate .

Statement 1:

Statement 2:

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.

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  and , making ; or, we can have   and , making . Neither scenario violates the conditions given.

### Example Question #2 : Dsq: Understanding Rays

, and  are distinct points.

True or false:  and  are opposite rays.

Statement 1:  is on

Statement 2:  is on

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.

EITHER statement ALONE is sufficient to answer the question.

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.

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

### Example Question #3 : Dsq: Understanding Rays

, and  are distinct points.

True or false:  and  are opposite rays.

Statement 1: .

Statement 2:  is the midpoint of .

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.

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.

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 .

In both figures, . But only in the second figure,  and  are on the opposite side of the line from , so only in the second figure,  and  are opposite rays.

Assume Statement 2 alone. If  is the midpoint of , then, as seen in the top figure,  is on . Therefore,  and  are the same ray, not opposite rays.

### Example Question #4 : Dsq: Understanding Rays

, and  are distinct points.

True or false:  and  are opposite rays.

Statement 1:

Statement 2:

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.

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 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.

,

thereby meeting the condition of Statement 1.

Also,  and  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,

.

Furthermore,  and  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  and  are opposite rays, then by segment addition, , making Statement 2 false. Contrapositively, if Statement 2 holds, and , then  and  are not opposite rays.

### Example Question #5 : Dsq: Understanding Rays

, and  are distinct points.

True or false:  and  are the same ray.

Statement 1: .

Statement 2:  is the midpoint of .

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 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.

Explanation:

We show Statement 1 alone is insufficient to determine whether the two rays are the same by looking at the figures below:

In both figures, , but only in the first figure,  and  are the same ray.

Assume Statement 2 alone. If  is the midpoint of ,  must be on , as in the top figure, so  and  are one and the same.

### Example Question #6 : Dsq: Understanding Rays

, and  are distinct points.

True or false:  and  are the same ray.

Statement 1:

Statement 2: .

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 insufficient 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.

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,

and

.

Also, since the three points are not on a single line,  and  are parts of different lines and cannot be the same ray.

Case 2:   with length 2 and midpoint .

and , so ; similarly, . Also,  and  are the same ray, since they have the same endpoint and  is on .

### Example Question #7 : Dsq: Understanding Rays

, and  are distinct points.

True or false:  and  are the same ray.

Statement 1: , and  are collinear.

Statement 2: .

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.

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:

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

Assume Statement 2 alone. Suppose  and  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,

,

Case 2: , and  are collinear.  must be between  and , as in the bottom figure, since if it were not,  and  would be the same ray. By segment addition,

,