Below is an image of a trapezoid
after it was reflected across the dotted line. Which of these figures shows how
the trapezoid appeared before the reflection?
The image on the left of the screen
shows how the trapezoid appears after the reflection. And we need to decide which of the
images (A), (B), (C), or (D) shows how the trapezoid appeared before.
A key property of reflections is
that they preserve the perpendicular distance of all points from the mirror
line. To determine how the trapezoid
appeared before the reflection, we can therefore consider the perpendicular distance
of each vertex of the trapezoid after the reflection and then find the point on the
opposite side of the mirror line at the same perpendicular distance. We note that the trapezoid has been
drawn on a squared grid and the mirror line is horizontal. Therefore, the perpendicular
distance of each point from the mirror line is its vertical distance.
Let’s consider each vertex of the
trapezoid. The pink vertex is three squares
above the mirror line. So its original position would have
been three squares below the mirror line at the same horizontal position. The red vertex is two squares below
the mirror line. So its original position would be
two squares above. The yellow and green vertices are
each two squares vertically from the mirror line. So, in fact, they have swapped
places under the reflection. Connecting the four crosses shows
how the trapezoid would’ve looked before the reflection took place.
Looking at the four answer options,
we can rule out options (A) and (D) as in both cases the sloping side below the
mirror line slopes upwards from left to right instead of downwards. Then, looking at option (B), we can
see that whilst the orientation is correct, this trapezoid is in the wrong position
on the grid. The bottom-left vertex should be
two squares below the mirror line, but it is only one square below. In option (C), however, each of the
vertices is in the same position as on our diagram. So option (C) shows how the
trapezoid appeared before the reflection.