# Question Video: Finding the Position of a Shape before a Reflection given Its Image Mathematics • 11th Grade

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? [A] Figure A [B] Figure B [C] Figure C [D] Figure D

02:29

### Video Transcript

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.