# Find the Main Reflecting Line of a Glide Reflection

A glide reflection involves three reflections, and so it can be challenging to find the location of its main reflecting line. Fortunately, there’s a handy theorem that you can use for just that purpose.

The main reflecting line of a glide reflection: In a glide reflection, the midpoints of all segments that connect pre-image points with their image points lie on the main reflecting line.

Ready for a glide reflection problem? This example will show you how to do the reflection first and then the translation, but you can do them in either order.

The following figure shows a pre-image parallelogram *ABCD* and the image parallelogram *A’B’C’D’* that resulted from a glide reflection. You job is to find the main reflecting line.

The main reflecting line in a glide reflection contains the midpoints of all segments that join pre-image points with their image points (such as *CC’*). You need only two such midpoints to find the equation of the main reflecting line (because you need just two points to determine a line). The midpoints of *AA’* and *BB’* will do the trick:

Now simply find the equation of the line determined by these two points:

Use this slope and one of the midpoints in the point-slope form and simplify:

That’s the main reflecting line. If you reflect parallelogram *ABCD* over this line, it’ll then be in the same orientation as parallelogram *A’B’C’D’* (*A* to *B* to *C* to *D* will be in the clockwise direction), and *ABCD* will be perfectly vertical like *A’B’C’D’.* Then a simple translation in the direction of the main reflecting line will bring *ABCD* to *A’B’C’D’*, as shown here.