Point Reflection

Task

The point in the $x$-$y$ plane with coordinates $(1000,2012)$ is reflected across the line $y=2000$. What are the coordinates of the reflected point?

IM Commentary

The purpose of this task is for students to apply a reflection to a single point. The standard 8.G.1 asks students to apply rigid motions to lines, line segments, and angles. Although this problem only applies a reflection to a single point, it has high cognitive demand if the students are prompted to supply a picture. This is because the coordinates of the point $(1000,2012)$ are very large. If students try to plot this point and the line of reflection on the usual $x$-$y$ coordinate grid, then either the graph will be too big or else the point will lie so close to the line of reflection that it is not clear whether or not it lies on this line. A good picture requires a careful choice of the appropriate region in the plane and the corresponding labels. Moreover, reflections of lines, line segments, and angles are all found by reflecting individual points.

This task was adapted from problem #3 on the 2012 American Mathematics Competition (AMC) 10B Test. The responses to multiple choice answers for the problem had the following distribution:

Of the 35,086 students who participated, 17,169 or 49% were in 10th grade, 9,928 or 28% were in 9th grade, and the remainder were below than 9th grade.

Solution

Below is a picture of part of the $x$-$y$ plane showing the point $(1000,2012)$ and its reflection over the line $y = 2000$:

The small boxes in the grid are each $1$ by $1$. Note that the axes in this picture are not the normal $x$-$y$ axes: the usual $x$-axis is the set of points where $y = 0$ and the usual $y$-axis is the set of points where $x = 0$. Here instead of the $x$-axis we look at the set of points where $y = 2000$ since this is the line of reflection for the problem. Instead of the usual $y$-axis, we have plotted the set of points where $x = 995$ so that our point $(1000,2012)$ will appear in the picture: any other choice of $x$ value sufficiently close to $1000$ would also work.

Note that the line of reflection $y = 2000$ is the perpendicular bisector of the blue segment. This means that the point $(1000,1988)$ is the reflection of $(1000,2012)$ about the line $y = 2000$.