Exercise 6.3.13
A) Prove that every point, P, on the perpendicular bisector of segment AB is the same euclidean distance from A as from B.
Proof: Let AB be a line segment with A = (a, 0) and B = (b, 0). Let D be the midpoint of AB such that D = (0, 0). Note that the y-axis is the perpendicular bisector of AB by definition. Furthermore, note that AD = DB. Pick some point on the perpendicular bisector of AB, say P = (0, p).
Using the Euclidean distance formula, we see that:
1) dist(A, P) = .
2) dist(B, P) = .
Note that since AD = DB, we know that .
Therefore, we know that dist(A, P) = dist(B, P) for any point P such that P is on the perpendicular bisector of AB.
B) Using the taxicab metric, it will hold true for line segments for which the slope, m, is either:
- m = 0
- m = 1
- m is undefined