Firstly, we are given that CM is a perpendicular bisector of segment AB at point M. This means that CM is a line that splits AB into two equal segments at a 90 degree angle. Therefore, point M is the midpoint of line segment AB.
In vector terms, we can describe our points as follows:
A(0,0) represents the initial point of our vectors.
B(b1,b2) represents the terminal point of our vectors.
Since M is the midpoint of AB, we will have AM = MB.
In vectors, the midpoint Formula is M = (A+B)/2
Substituting the coordinates of A and B into the midpoint formula:
M = ([0, 0] +[b1, b2])/2 = [b1/2, b2/2]
This result shows that the point M(b1/2, b2/2) is the midpoint of AB which confirms that AM = MB.
Given that CM is a perpendicular bisector of AB, it implies that AC = BC. This is because a perpendicular bisector of a segment is a line which cuts the segment into two equal parts at 90 degrees.
Therefore, AC = BC = Length of AM = Length of MB.
Thus, the lengths of segments AC and BC are equal.