# Equidistant to Points

- Author:
- mr. fay

## Definition: Equidistant

The

*set of*are the points that have equal distance to each element in the set S.**equidistant**points to a set S## Example: Vertices of a Pentagon

## 1. Draw all the points 4 units from the given point.

## 2. What is the set of all equidistant points to a given point? ( hint consider all distances )

## 3. Draw 4 points equidistant to the two given points.

4. What is the set of all equidistant points to the set of two distinct point?

## Proof

5. Prove the segments XA and XB are congruent. Recall corresponding parts of congruent triangles are congruent.

This implies that any point on the perpendicular bisector of 2 points is equidistant to those 2 points.
But are they the only points?

## Proof

6. Prove the angles YMA and BMY are congruent and therefore 90 degrees.

Therefore the point Y is on the perpendicular bisector.