When constructing a perpendicular bisector, we are specifically being asked to construct a line perpendicular to a line through the midpoint. To construct a perpendicular to a line through a point like the midpoint, we use a process similar to constructing a perpendicular to a line through a point not on the line. To construct a perpendicular, we use a compass and straightedge to determine a point equidistant from two equidistant points on the line.
When you're asked to construct a line perpendicular to a given line through a point on that line, it's going to be similar to the process of when you have a point not on the line and you're trying to find that perpendicular. So what do we have here, is we have a point that's on the line and we're trying to find that perpendicular that passes through p. So to do that I'm going to start off by making two points with my compass. So you're going to take your compass, it doesn't matter what the distance is that you have it's set apart, and I'm going to make a point over here and a point on the other side of point p. So now I've created two points that are the same distance from point p. So if I can find a point, either above or below point p, that is the same distance from both of these endpoints, then according to our definition that point would have to be on the perpendicular bisector. So I'm going to keep my compass and I'm going to swing an arc from one point here. I'm going to make my compass a little larger. So I'm going to swing an arc from this endpoint, swing another arc from over here and we have two points which is enough for a line. So I'm going to connect this point of intersection with the point that we started with and we have our, no we have an incomplete, it will be complete in a second. We have our perpendicular that passes through point p perpendicular to line l. This is a great technique when you're trying to construct a square or a rectangle in the construction problem.