Video Transcript
Given a circle with center 𝑀
that intersects with line 𝐿 at points 𝐴 and 𝐵, draw an image of circle 𝑀
after a reflection in line 𝐿. Which of the following
statements is correct? Is it option (A) line segment
𝐴𝑀 is parallel to line segment 𝐴𝑀 prime? Option (B) line segment 𝐵𝑀 is
parallel to line segment 𝐵𝑀 prime. Option (C) line segment 𝑀𝑀
prime is perpendicular to line segment 𝐴𝐵. Option (D) 𝑀𝑀 prime is equal
to 𝐴𝐵. Or option (E) 𝐴 prime 𝐵 prime
is greater than 𝐴𝐵.
To reflect a circle in a mirror
line, we must first reflect its center and then preserve its radius. We reflect the center, that’s
point 𝑀, by first creating the perpendicular line to 𝐿 that passes through
𝑀. Next, we know that line segment
𝑀𝐴 is a radius of our original circle. So, we can trace an arc with
center 𝐴 and with the same radius. The point where this arc
intersects our line is the center of our image. So, we have the image of our
circle after reflection. We can now use this to identify
the correct statement.
There’s no way that 𝐴𝑀 and
𝐴𝑀 prime can be parallel to one another. They quite clearly meet and
form an acute angle. In fact, 𝐵𝑀 and 𝐵𝑀 prime
cannot be parallel either for the same reasons. Of course, we do know that 𝑀𝑀
prime must be perpendicular to 𝐴𝐵. And this is because we
constructed the perpendicular line bisector of line 𝐿 at the very start. And line 𝐿 passes through
𝐴𝐵. The correct answer is option
(C). Line segment 𝑀𝑀 prime is
perpendicular to line segment 𝐴𝐵.
