- 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 as and as 1. Using the midpoint formula, we can determine that the midpoint is located at . 2. Consider a fourth point equidistant from and . Denote as . 3. We can determine the distance from to using the distance formula. 4. Similarly, we can determine the distance from to . 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.