DSQ: Understanding rays - GMAT Quantitative Reasoning
Card 0 of 7
Note: Figure NOT drawn to scale.
Evaluate 
.
Statement 1: 
Statement 2: 
Note: Figure NOT drawn to scale.
Evaluate
Statement 1:
Statement 2:
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.
Even with both statements,
For example, we can have
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, 
, and 
are distinct points.
True or false: 
and 
are the same ray.
Statement 1: 
Statement 2: 
.
True or false:
Statement 1:
Statement 2:
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 .
We show that both statements together provide insufficient information by giving two scenarios in which both statements are true.
Case 1:
Also, since the three points are not on a single line,
Case 2:
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, 
, and 
are distinct points.
True or false: 
and 
are the same ray.
Statement 1: 
, 
, and 
are collinear.
Statement 2: 
.
True or false:
Statement 1:
Statement 2:
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.
Statement 1 alone does not prove the rays to be the same or different, as seen in these diagrams:
In both figures,
Assume Statement 2 alone. Suppose
Case 1:
contradicting Statement 2.
Case 2:
contradicting Statement 2.
By contradiction,
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, 
, and 
are distinct points.
True or false: 
and 
are the same ray.
Statement 1: 
.
Statement 2: 
is the midpoint of 
.
True or false:
Statement 1:
Statement 2:
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 both figures,
Assume Statement 2 alone. If
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, 
, and 
are distinct points.
True or false: 
and 
are opposite rays.
Statement 1: 
.
Statement 2: 
is the midpoint of 
.
True or false:
Statement 1:
Statement 2:
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.
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,
In both figures,
Assume Statement 2 alone. If
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, 
, and 
are distinct points.
True or false: 
and 
are opposite rays.
Statement 1:
Statement 2:
True or false:
Statement 1:
Statement 2:
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.
Statement 1 alone does not answer the question.
Case 1: Examine the figure below.
thereby meeting the condition of Statement 1.
Also,
Case 2: Suppose
The three points are vertices of a triangle, and by the Triangle Inequality Theorem,
Furthermore,
Now assume Statement 2 alone. As can be seen in the diagram above, if
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, 
, and 
are distinct points.
True or false: 
and 
are opposite rays.
Statement 1: 
is on 
Statement 2: 
is on 
True or false:
Statement 1:
Statement 2:
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.
Both statements are equivalent, as both are equivalent to stating that
In both of the above figures,
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