Video Transcript
The following figure shows
rectangle 𝐴𝐵𝐶𝐷 drawn on a grid. Which of these could be the image
of the rectangle after it has been reflected across a horizontal line?
The image on the left of the screen
shows the rectangle before it is reflected across a horizontal line. And we’re asked to determine which
of the four options (A), (B), (C), or (D) could be its image after reflection. Note that we aren’t told exactly
where this horizontal line is. So we’re looking for an image that
is a reflection in any horizontal line. Let’s consider what happens to each
of the vertices of the rectangle when we reflect it in a horizontal line, which
we’ll arbitrarily choose to be in the vertical center of the squared grid.
A key property of reflection in a
line is that it preserves the perpendicular distance between all points and the
mirror line. In the case of a horizontal mirror
line, the perpendicular distance between each point and the mirror line will be
vertical. Let’s consider vertex 𝐴, which is
three squares vertically above the mirror line. Its image will be three squares
vertically below the mirror line at point 𝐴 prime. Point 𝐵 is also three squares
vertically above the mirror line, so its image 𝐵 prime will be three squares
vertically below. Points 𝐶 and 𝐷 are each one
square vertically above the mirror. So their respective images will
each be one square vertically below. Connecting the four points together
shows what one possible image of the rectangle after reflection in a horizontal line
would look like.
Now, remember we arbitrarily chose
a horizontal line to reflect in, so this isn’t the only possibility. But we can use this to deduce some
properties of what any reflection in a horizontal line would look like. First, the image is vertically
below the original. It hasn’t moved to the left or
right, so the image will be in the same horizontal position on the grid. Secondly, we can identify that the
vertices will be labeled in alphabetical order in a counterclockwise direction,
starting from the bottom left vertex. This relative orientation of the
vertices will be true regardless of the particular horizontal line we reflect the
rectangle in.
Now, looking at the four options,
we can rule options (C) and (D) out straightaway because they are in the wrong
position on the grid. They are actually in the position
corresponding to a reflection in a vertical line, although we would need to consider
the relative positioning of the vertices to determine which is correct.
If we consider the position of the
vertices for options (A) and (B), we see that the vertices are in the correct
position and order for option (B), whereas they are in the wrong position for option
(A). So option (B) is the correct
answer. The rectangle in this diagram could
be the image of the original rectangle after it has been reflected across a
horizontal line.
