Understanding rays - GMAT Quantitative
Card 1 of 112
In the above figure, give the union of 
and 
.
In the above figure, give the union of
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can be seen to be completely contained in 
- that is, 
. The union of a set and its subset is the containing set, so the correct response is 
.
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In the above figure, give the intersection of 
and 
.
In the above figure, give the intersection of
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can be seen to be completely contained in 
- that is, 
. The intersection of a set and its subset is the subset, so the correct response is 
.
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In the above figure, which two rays have as their union 
?
In the above figure, which two rays have as their union
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An angle, by definition, is the union of two rays with the same endpoint. Their common endpoint is the vertex of the angle, which is always named by the middle letter of a three-letter angle name. Therefore, we are looking for two rays with endpoint 
. Since the first letter of the name of a ray is always its endpoint, we are looking for two rays with 
as the first letter in their names. This makes 
and 
the correct choice.
An angle, by definition, is the union of two rays with the same endpoint. Their common endpoint is the vertex of the angle, which is always named by the middle letter of a three-letter angle name. Therefore, we are looking for two rays with endpoint
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Refer to the above figure. 
and which of the following are opposite rays?
Refer to the above figure.
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Opposite rays begin at the same endpoint; their directions are opposite each other. Since 
has endpoint 
, we are looking for the ray that has endpoint 
and goes in the opposite direction - this ray is 
.
Opposite rays begin at the same endpoint; their directions are opposite each other. Since
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Refer to the above figure. Which of the following is another name for 
?
Refer to the above figure. Which of the following is another name for
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A ray can be named with two letters, the first of which must be its endpoint and the second of which can be any other point on the ray.
has endpoint 
. The only other marked point on the ray is 
, so the correct choice is 
.
A ray can be named with two letters, the first of which must be its endpoint and the second of which can be any other point on the ray.
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A ray starts at the point 
and has a positive slope of 
. In which quadrants does some part of the ray lie?
A ray starts at the point
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A ray starts at a single point and then continues in a straight line infinitely in some direction. The given ray starts at the point 
, which is in quadrant 
, so we immediately know this must be included in the answer. The ray has a positive slope of 
, which means the next point is 
, followed by 
, 
, 
, 
, and so on. By plotting these points we can visualize that the ray starts in quadrant 
, crosses through quadrant 
, and then continues infinitely into quadrant 
, without ever crossing through any part of quadrant 
. The answer, therefore, is the following three quadrants:
A ray starts at a single point and then continues in a straight line infinitely in some direction. The given ray starts at the point
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Name a pair of opposite rays in the above figure.
Name a pair of opposite rays in the above figure.
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Opposite rays are rays that have the same endpoint are on the same line, but share only the endpoint.
Two names of opposite rays must begin with the same letter, since the first letter in the name of a ray is the endpoint. Also, the second letters must represent points collinear with each other and the endpoint, and on opposite sides of the shared endpoint. Of the five choices, only 
and 
together meet this criteria.
Opposite rays are rays that have the same endpoint are on the same line, but share only the endpoint.
Two names of opposite rays must begin with the same letter, since the first letter in the name of a ray is the endpoint. Also, the second letters must represent points collinear with each other and the endpoint, and on opposite sides of the shared endpoint. Of the five choices, only
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In the above figure, which of the following gives a correct name for the union of 
and 
?
In the above figure, which of the following gives a correct name for the union of
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and 
are noncollinear rays with the same endpoint, so their union is an angle. The shared endpoint of the rays, which here is 
, is the vertex of the formed angle, and is therefore the middle letter of the name of the angle; this letter is flanked by two letters representing one point on each ray, which are 
and 
. The correct choice among those given is 
.
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Which of the following could be another name for 
in the above figure?
I) 
II)
III) 
Which of the following could be another name for
I)
II)
III)
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A ray is named after its endpoint and any other point on the ray, in that order. Since 
is the first letter in the name 
, 
is its endpoint, and any other name for the ray must begin with 
; this allows us to eliminate 
. Also, 
is eliminated, since a ray is named after two, not three, points. 
is a correct choice, since the first letter in this name is endpoint 
, and the second letter names a point on this ray.
A ray is named after its endpoint and any other point on the ray, in that order. Since
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Give the union of 
and 
in the above figure.
Give the union of
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The diagram below shows 
and 
in red and green, respectively:
The union of the two sets is the set of points in one or the other; this set is the entire line containing the two rays, which is 
.
The diagram below shows
The union of the two sets is the set of points in one or the other; this set is the entire line containing the two rays, which is
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In the above figure, give the intersection of 
and 
.
In the above figure, give the intersection of
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The diagram below show 
and 
in red and green, respectively:
The intersection of 
and 
is the set of points they have in common, which can be seen to be the portion of the line with endpoints 
and 
. This figure is the line segment 
.
The diagram below show
The intersection of
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In the above diagram, let 
and 
be the midpoints of 
and 
, respectively, and 
and 
be the midpoints of 
and 
, respectively. Which of the following is not a valid alternative name for 
?
In the above diagram, let
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Below is the diagram with the points 
, and 
, as described, shown in green. Also, 
, the ray that has endpoint 
and passes through 
, is marked in red.
The ray also passes through 
and 
, so 
, 
, 
, and 
—all four given names—are also valid names for the ray.
Below is the diagram with the points
The ray also passes through
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In the above diagram, let 
and 
be the midpoints of 
and 
, respectively, and 
and 
be the midpoints of 
and 
, respectively. Which of the following is not a valid alternative name for 
?
In the above diagram, let
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Below is the diagram with the points 
, and 
, as described, shown in green. Also, 
, the ray that has endpoint 
and passes through 
, is marked in red.
The ray also passes through 
, 
, and 
, so 
, 
, and 
are also valid names for the ray. However, the ray does not pass through 
, so 
is not a valid name for the ray. 
is the correct choice.
Below is the diagram with the points
The ray also passes through
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In the above figure, let 
be the midpoint of 
. Which of the following would give another name for 
?
In the above figure, let
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Below is the diagram with midpoint 
of 
added; also, , the ray starting at 
and passing through 
, is in green.
A ray is named after, in order, its endpoint and any other point on the ray. 
has 
as an endpoint, and also includes the points 
and 
, so there are two valid alternative names, 
and 
, among the choices. The correct response is .
Below is the diagram with midpoint
A ray is named after, in order, its endpoint and any other point on the ray.
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In the above figure, give the union of and .
In the above figure, give the union of
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can be seen to be completely contained in - that is, . The union of a set and its subset is the containing set, so the correct response is .
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In the above figure, give the intersection of and .
In the above figure, give the intersection of
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can be seen to be completely contained in - that is, . The intersection of a set and its subset is the subset, so the correct response is .
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In the above figure, which two rays have as their union ?
In the above figure, which two rays have as their union
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An angle, by definition, is the union of two rays with the same endpoint. Their common endpoint is the vertex of the angle, which is always named by the middle letter of a three-letter angle name. Therefore, we are looking for two rays with endpoint . Since the first letter of the name of a ray is always its endpoint, we are looking for two rays with as the first letter in their names. This makes and the correct choice.
An angle, by definition, is the union of two rays with the same endpoint. Their common endpoint is the vertex of the angle, which is always named by the middle letter of a three-letter angle name. Therefore, we are looking for two rays with endpoint
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Refer to the above figure. and which of the following are opposite rays?
Refer to the above figure.
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Opposite rays begin at the same endpoint; their directions are opposite each other. Since has endpoint , we are looking for the ray that has endpoint and goes in the opposite direction - this ray is .
Opposite rays begin at the same endpoint; their directions are opposite each other. Since
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Refer to the above figure. Which of the following is another name for ?
Refer to the above figure. Which of the following is another name for
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A ray can be named with two letters, the first of which must be its endpoint and the second of which can be any other point on the ray.
has endpoint . The only other marked point on the ray is , so the correct choice is .
A ray can be named with two letters, the first of which must be its endpoint and the second of which can be any other point on the ray.
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A ray starts at the point and has a positive slope of . In which quadrants does some part of the ray lie?
A ray starts at the point
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A ray starts at a single point and then continues in a straight line infinitely in some direction. The given ray starts at the point , which is in quadrant , so we immediately know this must be included in the answer. The ray has a positive slope of , which means the next point is , followed by , , , , and so on. By plotting these points we can visualize that the ray starts in quadrant , crosses through quadrant , and then continues infinitely into quadrant , without ever crossing through any part of quadrant . The answer, therefore, is the following three quadrants:
A ray starts at a single point and then continues in a straight line infinitely in some direction. The given ray starts at the point
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