Proof 5.12

Author:: Kayla Moore

Using coordinates, write a detailed step-by-step proof that the set of points equidistant from the two fixed points A and B is the perpendicular bisector of the segment AB.

Let

be a line segment with end points

and

. Denote

and

1. Using the midpoint formula, we can determine that the midpoint

is located at

. 2. Consider a fourth point

equidistant from

and

. Denote

. 3. We can determine the distance from

using the distance formula.

4. Similarly, we can determine the distance from

5. Since we know that

is equidistant from

and

, we can set the formulas from 3 and 4 equal to one another and solve for

. 6. Since the x-coordinate of

is zero, we know that it lies on y-axis. Recall,

also lies on the y-axis which is perpendicular by definition to the x-axis where

and

are located. 7. Since

is the midpoint of

and lies on a line perpendicular to

, the line

is the perpendicular bisector of

. Therefore, the set of points equidistant from two fixed points is the perpendicular bisector of the segment formed by the fixed points.

Proof 5.12

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