Definition: A perpendicular bisector of a segment is a line that passes through its midpoint at a right angle; each of its points is equidistant from the endpoints.
What are its two properties?
It cuts the segment into two equal parts, and it meets the segment at 90 degrees. The perpendicular bisector theorem says that any point on it is the same distance from both endpoints. The converse also holds.
How do you write its equation?
Find the midpoint of the segment. Find the slope of the segment, then take the negative reciprocal. Use point-slope form.
Example. Segment AB has A(1, 2) and B(7, 6). The midpoint is (4, 4). The slope of AB is 4/6 = 2/3, so the perpendicular slope is -3/2. The equation is y – 4 = -(3/2)(x – 4), which simplifies to y = -(3/2)x + 10.
What mistakes occur?
Using the slope of AB instead of its negative reciprocal, which would give a parallel line, or forgetting the midpoint and using an endpoint. A line that is perpendicular but not through the midpoint is not a bisector. The three perpendicular bisectors of a triangle meet at the circumcenter.