Proof: 5.2
Proof: 5.2
Using coordinates, write a detailed step-by-step proof that the set of points equidistant from two fixed points, A and B, is the perpendicular bisector of the segment AB.
Let AB be a line segment with endpoints A and B. Let C be the midpoint of AB such that C is centered at the origin. Thus, we know that C = (0, 0). We will denote A and B as follows: A = (x, 0) and B = (-x, 0).
Now, let D be another point which is equidistant from line segment AB. Denote D as .
Using the distance formula, we see that:
1)
2)
Since we know that D is equidistance from both A and B, we can say that dist(A,D) = dist(B,D).
Consider the following:
.
Thus, for any point D that we choose equidistant from A and B, the x coordinate must always be zero. We know that C is the midpoint of AB, and the x coordinate of C is zero, thus lying on the y-axis. Note that by definition, the y-axis is perpendicular to the x-axis. Thus, any point D equidistant from A and B forms the line CD which is perpendicular to AB since point points lie on the y-axis. Therefore, the set of all points equidistant from AB must lie on line CD, the perpendicular bisector of AB.