Video Transcript
What does the following figure
illustrate? (A) A bisector of a line
segment. (B) A straight line parallel to
another line. (C) A bisector of an angle. (D) A perpendicular to a straight
line originating from it. Or (E) a perpendicular from a point
lying outside a straight line.
In the figure, we can observe that
we have a line passing through 𝐴 and 𝐵. And specifically we can see the
portion of this line which could be defined as the line segment 𝐴𝐵. We also have a point 𝐶, which lies
on the line. And we can further see that we have
arcs drawn on the diagram, which would indicate that some construction has been done
to create the ray from point 𝐶.
Now let’s consider the options that
we were given. Straightaway, we can eliminate
option (C), because no angles have been shown, either bisected or not. And we can also eliminate option
(B), because there are no parallel lines in the figure. We may consider option (A) to be a
possible answer. This does look quite similar to a
bisector or perpendicular bisector of a line segment.
However, let’s recall that the
perpendicular bisector of a line segment would look something like this. Notice that there are arcs above
and below the line segment that we use for constructing the bisector 𝐶𝐷 of a line
segment 𝐴𝐵. If there are arcs on just one side
of the line segment, then this would not be the construction of a perpendicular
bisector. So we can eliminate answer option
(A).
The final two options, (D) and (E),
both consider perpendiculars from a line, with 𝐷 concerning a perpendicular from a
point on the line and 𝐸 concerning a point outside the line. And if we consider the diagram, it
would appear that the point 𝐶 lies on the line. And this is the perpendicular from
this point. But in order to check that this is
the correct construction, let’s remind ourselves of the steps we would take to carry
out the construction.
We can draw a similar line 𝐴𝐵,
which contains a point 𝐶. And as always, if we’re
constructing, we’ll need to use a compass, which is one of these tools. The first step we take is that we
set our compass at point 𝐶 and trace a circle intersecting the line segment 𝐴𝐵
twice. That means that we place the
pointed end of the compass onto point 𝐶. We trace an arc crossing the line
on one side and another arc on the other side of point 𝐶. We can label the two points where
the arcs and line segment intersect as 𝐴 prime and 𝐵 prime.
Now, we will use our compass
again. This time, we want to set the
compass so that it’s larger than the length of 𝐴 prime 𝐶. And we trace arcs of two circles
centered at 𝐵 prime and 𝐴 prime such that they intersect. So, starting at 𝐴 prime, we draw
an arc, like this. And we don’t need to draw an arc
below the line segment as well like we would for a perpendicular bisector of a
line.
Doing the same with the compass
point on point 𝐵 prime this time, we would create the following pair of arcs above
the line. We can label the intersection of
these two arcs with the letter 𝐷. And joining 𝐶 and 𝐷, either with
a ray from 𝐶 through 𝐷 like this or a line segment between them, we have created a
perpendicular from the point 𝐶 on the line such that the line segment 𝐶𝐷 is
perpendicular to line segment 𝐴𝐵. And therefore the illustration
shown is a perpendicular to a straight line originating from it.
As an aside, if we had to draw a
perpendicular from a point lying outside a line segment, then it would look like
this given diagram. As this was not the same as the
diagram in the question, then we can also eliminate this option, leaving us with the
answer given in option (D).
