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Constructing the Perpendicular Bisector - Problem 1
Extend the compass to over half the length of the line segment (this can be done just by looking at it). Draw an arc through the line, starting at one endpoint. Repeat with the other endpoint, using the same measure of arc. Then, using the straightedge, connect the points at which the arcs intersect. This new segment is a perpendicular bisector of the given line segment, meaning that it intersects the line segment at its midpoint, forming right angles.
When you use a compass and a straightedge to find the midpoint of a line segment, you’re also finding the perpendicular bisector of that line segment. So I wrote this problem to say, find the midpoint and construct the perpendicular bisector. In essence you’re doing the same thing on one step.
But how do we construct a perpendicular bisector? Well, in order to draw a line, we need two points. Two points make a line that’s the definition that we’ve known ever since back in grade school.
So I’m going to take my compass and I’m going to extend the width of the compass to more than half of the line segment. Again that’s key, if you just had your compass really small, it’s not going to work. So I’m going to go to more than half of the line segment and I’m going to swing an arc from one endpoint. So I’m going to go to point A and I’m going to swing an arc from point A.
So these are all the points that are a certain distance away from A. without changing my compass, I’m going to go over to point B and I’m going to do the same thing, I’m going to swing another arc so that way I can see where those two points intersect. So at this point right here and this point right here we have a point that is the same distance away from A as it is away from B which is the definition of a perpendicular bisector.
So I’m going to connect these two points using my straightedge. And by doing this I am creating a perpendicular bisector and when I said bisector that means that this point right here is the midpoint of line segment AB thereby creating two congruent line segments.
So if you want to create the perpendicular bisector or the midpoint, you need to swing an arc from your two end points that’s exactly identical. So you have two points of intersection which have to be on your perpendicular bisector.