Video Transcript
How can the ray shown in the figure
be represented mathematically? Options (A), (B), (C), (D), and
(E).
In this question, we are given a
ray and two distinct points on the ray and asked to identify which of five options
correctly represents the ray mathematically. To do this, let’s start by
recalling what is meant by a ray.
We can recall that a ray is a
straight line with a single endpoint that extends indefinitely in one direction. In the diagram, we can see that the
point 𝐴 is the endpoint of the ray, that is, where the ray starts. We can also see that the ray
extends indefinitely in the direction from 𝐴 through 𝐵. The arrow shows that the line
continues indefinitely in that direction.
To represent this object
mathematically, we use an arrow over the two points. It is important that the arrow
starts at 𝐴 to show that 𝐴 is the endpoint. The arrow’s direction must match
the direction the ray extends indefinitely. We can see that this is given by
option (B).
There is something worth noting
about this notation. This notation means the set of all
points that lie on the ray. This means that the notation for a
ray is not unique. For instance, if 𝐶 is another
point on the ray, then the ray from 𝐴 through 𝐵 is the same as the ray from 𝐴
through 𝐶.
For due diligence, we can also
recall what each of the other four options represents. First, 𝐴𝐵 is the distance between
𝐴 and 𝐵. Next, if the arrow starts at 𝐵 and
points at 𝐴, then this is the ray from 𝐵 through 𝐴. So it is the ray that extends in
the other direction. Next, if we have a horizontal line
over 𝐴 and 𝐵, then our line does not extend indefinitely in either direction. This is called a line segment, and
𝐴 and 𝐵 are its endpoints. Finally, if we have a double-sided
arrow over 𝐴 and 𝐵, then this is the line passing through 𝐴 and 𝐵. And it extends indefinitely in both
directions.
Hence, only option (B) represents
the ray from 𝐴 through 𝐵 mathematically.
