In the previous post, we showed that all of the points on the perpendicular bisector of a segment are equidistant from the endpoints of the segment. But are these the only points with this equidistance property? Let’s use the properties of reflections to show that they are.
Let segment be given, and let be a point in the plane that is equidistant from and . To show that is on the perpendicular bisector of , we’ll find a line through in which is the mirror image of . That is, we’ll show that is the image of under the transformation that reflects the plane over line .
First let’s construct a circle through centered at . Since and are the same distance from , this circle passes through as well as .
Which line through will act as a mirror for the endpoints of ?
Click here to interact with the figure in GeoGebra.
The entire figure appears to be line-symmetric, but we need some way of describing the line of symmetry. To that end, let be the point that divides the arc from to into two equal parts, and let be the line through and . Let be the reflection of the plane over line – we’ll show that maps onto .
Let . To show that , we need to answer two questions: How can we be sure that is on the same circle as and ? How can we be sure that is located precisely at point on this circle?
- Since and reflections preserve distance, and must be equidistant from . That is, and are on the same circle centered at .
- Let represent the measure of arcs and . Since , , and reflections preserve angles, the measure of must be as well. Since there is only one counter-clockwise arc from with degree measure , it follows that coincides with .
- Having shown that , it follows from the definition of reflection that line is the perpendicular bisector of segment . But point is on , which means that is on the perpendicular bisector of – this was to be proved!
This proof relies on three very important ideas:
- The definition of reflection
- The fundamental assumption that reflections preserve distance
- The fundamental assumption that reflections preserve angles
We are now in a position to completely characterize the points on the perpendicular bisector of a segment, thus fulfilling the final requirement of Standard G-CO.9:
The points on the perpendicular bisector of a segment are exactly those that are equidistant from the segment’s endpoints.
This theorem has many applications. In particular, it will play a critical role in our proof of the Side-Side-Side criterion for triangle congruence.