### All GMAT Math Resources

## Example Questions

### Example Question #1 : Understanding Rays

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

**Possible Answers:**

**Correct answer:**

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 #1 : Understanding Rays

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

**Possible Answers:**

None of the other choices is correct.

**Correct answer:**

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?

**Possible Answers:**

**Correct answer:**

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 ?

**Possible Answers:**

and

and

and

and

and

**Correct answer:**

and

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 .

**Possible Answers:**

**Correct answer:**

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 .

**Possible Answers:**

**Correct answer:**

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)

**Possible Answers:**

I only

II only

I and III only

I, II, and III

I and II only

**Correct answer:**

I only

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 #1 : 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 ?

**Possible Answers:**

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

**Correct answer:**

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.

**Possible Answers:**

and

and

and

and

and

**Correct answer:**

and

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 #1 : Understanding Rays

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

**Possible Answers:**

**Correct answer:**

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 .