For any line  with an equation  and slope , a line that is perpendicular to  must have a slope of , or the negative reciprocal of . Given the equation , we know that  and therefore know that . 

Given a slope of , we know that there are two solutions provided:  and . 

First, we need to rearrange the equation into slope-intercept form.  .

  Therefore, the slope of this line equals  Perpendicular lines have slope that are the opposite reciprocal, or 

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  .

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 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:

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.

 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 .

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 .

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.

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.