DSQ: Understanding rays - GMAT Quantitative

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 .

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,

,

contradicting Statement 2.

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,

,

contradicting Statement 2.

By contradiction, and are the same ray.

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.

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.

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.

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.

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.

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 .

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,

,

contradicting Statement 2.

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,

,

contradicting Statement 2.

By contradiction, and are the same ray.

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.

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.

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.

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.

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.

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 .

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,

,

contradicting Statement 2.

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,

,

contradicting Statement 2.

By contradiction, and are the same ray.

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.

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.

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.