# GMAT Math : Understanding rays

## Example Questions

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### Example Question #1 : Understanding Rays

Refer to the above figure.  and which of the following are opposite rays?

Explanation:

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  .

### Example Question #2 : Understanding Rays

Refer to the above figure. Which of the following is another name for  ?

None of the other choices is correct.

Explanation:

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 .

### Example Question #3 : Understanding Rays

A ray starts at the point    and has a positive slope of  .  In which quadrants does some part of the ray lie?

Explanation:

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:

### Example Question #4 : Understanding Rays

In the above figure, which two rays have as their union  ?

and

and

and

and

and

and

Explanation:

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.

### Example Question #5 : Understanding Rays

In the above figure, give the intersection of  and .

Explanation:

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 .

### Example Question #6 : Understanding Rays

In the above figure, give the intersection of  and .

Explanation:

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 .

### Example Question #7 : Understanding Rays

Which of the following could be another name for  in the above figure?

I)

II)

III)

I, II, and III

II only

I and III only

I only

I and II only

I only

Explanation:

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.

### Example Question #8 : Understanding Rays

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  ?

Each of the other choices gives a valid alternative name for .

Explanation:

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.

### Example Question #9 : Understanding Rays

Name a pair of opposite rays in the above figure.

and

and

and

and

and

and

Explanation:

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.

### Example Question #10 : Understanding Rays

In the above figure, which of the following gives a correct name for the union of  and ?