Video Transcript
True or False: In the following
figure, line 𝐶𝐷 is a line of symmetry of line segment 𝐴𝐵.
Let’s begin by considering the
figure and the line segment 𝐴𝐵. We can observe in the figure that
there are two pairs of arcs above and below the line segment. These arcs have been created by
using a compass, which is a tool that we use when we are creating geometrical
constructions. For example, in this construction,
we began with the line segment 𝐴𝐵. The compass was placed either on
point 𝐴 or 𝐵 first; it doesn’t matter which. Here, we have placed it on point
𝐵. And arcs were drawn above and below
the line segment like this. Then the compass point was placed
on the other endpoint of the line segment. And a pair of arcs were drawn above
and below the line segment again. The points of intersection of the
arcs above and below were then joined with a line.
By doing so, we have created the
perpendicular bisector of the line segment 𝐴𝐵. This is also demonstrated in the
original diagram. We can observe that there are two
congruent line segments such that 𝐴𝐵 has been bisected. And because of these important
arcs, we know that this is a perpendicular bisector and therefore the line segments
𝐴𝐵 and 𝐶𝐷 must be perpendicular.
We can now consider the problem, is
the line 𝐶𝐷 a line of symmetry of line segment 𝐴𝐵? And of course, it must be, because
we know that we have these congruent line segments as 𝐴𝐵 is bisected. If we reflected the left side onto
the right side across the line 𝐶𝐷 or vice versa, then the two sections would fit
exactly on top of one another.
So, the answer to the problem is
true. Line 𝐶𝐷 is a line of symmetry of
line segment 𝐴𝐵.
