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Constructing the Perpendicular Bisector - Concept
When looking at a line segment, there is only one line that will pass through the midpoint that will be a constant distance between the two endpoints. This line is called the perpendicular bisector. To construct the perpendicular bisector, we first find the midpoint of the line segment and then use a compass and straightedge to draw the perpendicular line.
If we look at this line segment ab and we're talking about a midpoint there's only going to be one line that will pass through this midpoint that would be a constant distance from a and b. So to give a little contrast, let's say I drew a line kind of like that, it's pretty clear to see that if you're at a point up here you're going to be closer to point a and if you're at a point down here you're going to be closer to point b.
What we want to do is we want to construct a line through this point, the midpoint that will always be the exact same distance from a and b. So the way that we're going to do that, is we're going to use our compass and straight edge our tools of construction and we're going to make a perpendicular line. So this line will intersect line segment ab at a 90 degree angle and it will divide ab into two congruent pieces. So notice that if you picked any point along this perpendicular it will be the same distance from the two end points. So that's kind of a definition of a perpendicular bisector. It's a collection of all the points that are the same distance from the two end points of a line segment.