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Locus and Definition of a Circle and Sphere - Problem 1
The locus is the set of points that meet a certain condition. A circle is defined as the locus of points that are a certain distance from a given point.
So, given a line segment and its endpoints, the locus is the set of points that is the same distance from both endpoints. From the definition of a midpoint, the midpoint is equidistant from both endpoints. In addition, each point on a perpendicular bisector of this segment has the same distance between each endpoint (which is not the same distance as the other points on the perpendicular bisector). So, the perpendicular bisector of a line segment is its locus.
In this question, we’re asked to apply what we know about a locus to a new situation. It says draw a locus of points in this plane, the plane on the board, that are equidistant from A and B.
Well, let’s start off by saying, what is a locus? Well locus is the set of points that meet a given condition. We said the definition of a circle is the locus of points that are a given distance from a point that’s given to you.
So if I look at this line segment, I know one point that will be equidistant from these two is the midpoint. So I can mark that point right there which is going to be the midpoint of that line segment. But that’s not the only one. There’s also going to be a point up here that will be equidistant from both of those endpoints. So that’s also going to be one down below that will be equidistant from both of those endpoints.
So basically when it’s asking for the locus, we’re asking for a line in this problem and that line happens to be the perpendicular bisector. By definition, every single one of these points on this line is equidistant from A and from B. So this line represents the locus of points or all the points that are equidistant from A and from B.