Video Transcript
Which of the following
constructions represents drawing the line of symmetry, perpendicular bisector, of
line segment 𝐴𝐵?
In this problem, we need to recall
how we create the perpendicular bisector of a line segment between points 𝐴 and
𝐵. A perpendicular bisector will
always create a line of symmetry of a line segment. We can note that the terminology of
bisector means to cut something into two equal pieces or parts. And perpendicular means at right
angles. So our finished perpendicular
bisector will look something like this.
However, when we are constructing a
perpendicular bisector, we can’t just find the halfway point on a line segment and
draw a 90-degree line. We must follow a particular
process. So let’s clear the answer options
for a bit while we see how we carry out the construction of a perpendicular bisector
of a line segment.
Here, we have a line segment 𝐴𝐵
drawn, and it can be any length. But it’s useful to leave some space
above and below the line. Or if we have a vertical line
segment, we should leave some space to the right and left of the line segment. Now, because we are constructing,
we need this tool, which is called a compass, or sometimes a pair of compasses,
depending on where you live.
The first step when we are
constructing a perpendicular bisector is to set the radius of the compass to be
greater than half the length of the line segment 𝐴𝐵. That means we’ll have to adjust the
legs of the compass to be an appropriate length. And if the legs are moving really
easily, then it’s always a good idea to make sure the screw at the top is tightened
so that they don’t move about when we are drawing circles or arcs.
The next step is to trace arcs of
two circles centered at 𝐴 and 𝐵. And we can start with point 𝐴. That means that we should move our
compass and place the pointed end onto point 𝐴. We then draw an arc above the line
segment and below the line segment. We can draw the entire circle about
point 𝐴 if we wish. But often it’s easier to see the
finished diagram if we just draw two arcs like this. So that’s the arcs centered at 𝐴
done.
We then need to do the same for the
arcs centered at 𝐵. To do this, we lift the compass,
turn it around, and put the pointed end on point 𝐵. And we draw arcs above and below
the line segment like this. When we have completed this, we
observe that each pair of arcs above and below the line segment intersect. We can label these two points 𝐶
and 𝐷.
Finally, by joining the points 𝐶
and 𝐷, we create the perpendicular bisector 𝐶𝐷 of the line segment 𝐴𝐵. Because line segment 𝐴𝐵 is
bisected, we know that it is split into two congruent pieces. And this is done at right
angles. Following these steps will allow us
to construct the perpendicular bisector of any line segment. And it’s worth noting that we
should keep the arcs that we’ve drawn rather than erasing them, as these are a
demonstration that we have carried out the method correctly. But let’s return to the answer
options.
We can include the figure that we
created. And so we can see that it is the
answer given in option (C) that represents the construction of a perpendicular
bisector of a line segment 𝐴𝐵. Note that although the diagram
given in (B) does appear to show a perpendicular bisector, it doesn’t show the arcs
that we would need to conclude that a construction has been accurately carried
out. Therefore, option (C) is the only
correct answer for both the line of symmetry and the perpendicular bisector of line
segment 𝐴𝐵.
