Finding a Locus of Points from a Description
Date: 03/30/2004 at 12:35:54 From: Rene Subject: locus in geometry I am trying to help my son figure out these three questions. We have asked several people and no one can help us. Is there a formula to use? I know nothing when it comes to geometry, so hope you can help. They all have to do with the locus in geometry. 1. What is the locus of points in the plane of an angle that are equidistant from the sides of angle? 2. What is the locus of points in space that are equidistant from two parallel planes? 3. What is the locus of points in a plane that are equidistant from points A and B in the plane?
Date: 03/30/2004 at 13:03:47 From: Doctor Peterson Subject: Re: locus in geometry Hi, Rene. This isn't a matter of formulas; I'm not sure exactly what is the key to recognizing a locus. Each problem is a little different. In the hardest cases, you have to use analytic geometry, and only when you get an equation will you see what the locus actually is. In many cases, you just "see" it; the answer is obvious when you look at it the right way. These problems are in between; they're obvious to me because I know enough geometry, but you can figure them out visually or analytically if you don't immediately see it. First I'd like to point you to some discussions of what locus is, and some examples: Locus http://mathforum.org/library/drmath/view/55121.html The Meaning of Locus http://mathforum.org/library/drmath/view/62350.html Locus and Equations of Lines http://mathforum.org/library/drmath/view/52853.html Let's start your first problem: 1. What is the locus of points in the plane of an angle that are equidistant from the sides of angle? I would start by drawing a picture: / + P / o / | o-----+-- O I've drawn an angle, and a point P that is equidistant from the sides of the angle (that is, the same distance from each ray). I drew in (or would if I could draw!) perpendicular segments from P to each ray, to remind myself how to find the distance from a point to a line (or ray). The locus is not yet visible; it will be the set of all places where P might be ("locus" is just the Latin word for "place"), or equivalently the set of all points P that fit the definition. Now what you do next will depend on your thinking style. If you can visualize easily, imagine moving P around while keeping the two distances the same; you'll find that it slides along in a particular direction, tracing out a certain line. If you prefer experimenting with reality rather than picturing things in your mind, try finding several points that fit the description, and look for what they have in common. Then you can fill in between, to locate all possible points that work. On the other hand, if you like theory and have an analytical mind, you can just look for a theorem or two that will determine where P can be; in this case, try drawing in segment OP and you'll notice a pair of congruent triangles. What do they tell you about the location of P? Whatever method you use to discover the locus, or just guess at it, you need to convince yourself that your answer is correct, either with a real proof, or just enough thinking (along the lines of the analytical approach I listed last) to feel confident that you've got it right. So it takes a certain amount of "playing" and imagining to find the answer, and a certain amount of familiarity with theorems or facts of geometry to see that you have it right; locus problems test your knowledge and a bit more. But usually if you just think about the meaning of the terms in the problem (what is the distance from a point in space to a plane, for example?) and think concretely, you should be able to get it. If you need more help, please write back and show me how far you got on your problems. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 04/05/2004 at 11:33:22 From: Rene Subject: Thank you (locus in geometry) Thank you for your prompt and excellent help! We thought about the questions and hopefully got them right.
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