LocusDate: 05/03/99 at 08:37:49 From: Jo Subject: Locus What do you have to do to get the perpendicular bisector of all the loci of a triangle? What does it mean when it asks you about equiddistance from certain points? Thanks. Date: 05/03/99 at 13:02:03 From: Doctor Peterson Subject: Re: Locus Hi, Jo. Thanks for writing. Your first question doesn't make a lot of sense, because there is no "locus of a triangle." But I think I can see what you're asking about. A "locus" is the set of all points that satisfy some rule or description, and a perpendicular bisector is one of the simplest examples of a locus. If I asked you "what is the locus of all points equidistant from two given points?", the answer would be "the perpendicular bisector of the segment determined by the two points." Here's what the question means: What is the locus of all points ... (What geometrical object consists of all points X ...) that are equidistant from A and B? (... for which the distances AX and BX are equal?) That is, "equidistant from two points" means "the same distance from both points." Here's what the answer means: The perpendicular bisector ... (the line through the midpoint of the segment, perpendicular to the segment) of the segment determined by A and B. (you make the segment by connecting A and B with a straight line) You can prove (and probably your text did this) that any point on the perpendicular bisector of a segment AB is the same distance from both endpoints; you can see this by drawing in the segments and seeing that you have an isosceles triangle, so that AX = BX: | + X /|\ / | \ / | \ / | \ / | \ / | \ / | \ / | \ / | \ / | \ / | \ / | \ / |_ \ / | | \ +--------------+--------------+ A | B | | | | | | | This tells us that any point on the perpendicular bisector is equidistant from A and B, so it is part of the locus we are looking for. Conversely, you can show that if any point is equidistant from A and B, it must be on the perpendicular bisector, so there are no other points in the locus: the perpendicular bisector IS the entire locus we're looking for. That's what a locus is: the set of ALL points that fit some description (in this case, being equidistant from A and B). In our case, the perpendicular bisector is "the locus, the whole locus, and nothing but the locus" of equidistant points. There are many other examples of a locus. For example, the locus of points 1 inch from point A is the circle of radius 1 centered at A. Every point of the circle is 1 inch from A, and every point one inch from A is part of the circle. I hope that helps a little to clarify the idea of a locus, and the meaning of equidistant. If you have problems you still have trouble with, maybe you can write back with a specific problem so I can see the wording of it. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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