Video Transcript
Which of the following represents
the image of line segment 𝐴𝐵 after a reflection in line 𝑀?
We’ve been given five diagrams that
each show line segment 𝐴𝐵 and line 𝑀. We need to determine which of the
five diagrams shows the correct image of line segment 𝐴𝐵 after a reflection in
line 𝑀.
One of the key properties of
reflections in a line is that they preserve the perpendicular distance of all points
from the mirror line. So, when we’re looking for the
image of line segment 𝐴𝐵, it should be on the opposite side of the mirror line but
at the same perpendicular distance away. In fact, the mirror line is the
perpendicular bisector of the line segment connecting each point and its image.
Let’s look at diagram (A). Taking the diagonal of the smaller
squares on the grid to be our unit of measure, we can see that points 𝐴 and 𝐴
prime are each at a perpendicular distance of two and a half units from the mirror
line. So this looks good so far. But when we consider points 𝐵 and
𝐵 prime, they’re at perpendicular distances of 1.5 and 2.5 units from the mirror
line. So the perpendicular distances are
not the same. We can therefore rule out option
(A).
In option (B), the images of points
𝐴 and 𝐵 are in the wrong place. The images appear to have been
labeled the wrong way round. Line segments 𝐴𝐴 prime and 𝐵𝐵
prime are not perpendicular to the mirror line. So this figure can’t represent a
reflection of line segment 𝐴𝐵 in line 𝑀.
In option (C), points 𝐴 and 𝐴
prime are the same perpendicular distance from the mirror line, but points 𝐵 and 𝐵
prime aren’t. So this isn’t the correct image
either.
In option (D), the perpendicular
distances from the mirror line are the same for points 𝐴 and 𝐴 prime, which are
each 2.5 units, and for points 𝐵 and 𝐵 prime, which are each 1.5 units. So option (D) does show the correct
image of line segment 𝐴𝐵 after a reflection in line 𝑀.
For due diligence, we should check
the perpendicular distances in option (E). We may be able to judge by eye that
they are not the same. But if we add the distances to the
diagram, this confirms it.
The correct figure which shows the
image of line segment 𝐴𝐵 after a reflection in line 𝑀 is option (D).
