The Centroid of a TriangleDate: 02/25/98 at 00:20:39 From: Pauline Jamry Subject: WHY is the centroid of any triangle the center of its balance? WHY is the centroid of any triangle the center of its balance? I have searched your Web site and have found how to find a centroid and that it is the center of a triangle's balance, but not why. Is there a formula or theorem? I know how to find the centroid of a triangle by taking the intersection of its medians. There is an easier and much quicker algebraic way of finding the centroid of a triangle, but I couldn't find that. I thought the reason why the centroid was the balance point was that if you take half of any line of a triangle, i.e., its midpoint, and just connect it to the opposite vertex, this median and the other two of the triangle will be close enough reason to the truth, but I need basis and facts. I also thought that the algebraic way of finding the centroid would help me answer my question, or at least give me a clue. Date: 02/25/98 at 08:44:38 From: Doctor Jerry Subject: Re: WHY is the centroid of any triangle the center of its balance? Hi Pauline, Let me first say that, for me, the most utterly convincing arguments for calculating centroids or, more generally, centers of mass, reside in calculus. I'm judging that you are not seeking an answer depending upon calculus. Here is a classical argument as to why the centroid of a triangle is 2/3 of the way from vertex to midpoint of opposite side. The argument has two parts. 1. Convince yourself that the medians do in fact intersect at a point that is 2/3 of the way from the vertex. This is a question independent of whether this point is the centroid or balance point. I'm assuming that you can do this. 2. Now take a triangle and draw one median, from vertex C to side c. Draw three or four representative lines through the median, parallel to side c. Each of these lines is bisected by the median. This is a Euclidean geometry exercise. So, thinking about the triangle as made from an infinite number of these parallel lines, it appears that the balance point must be on the median, since it bisects each of these lines. Half the mass (line-by-line) is on one side of the median and half the mass is on the other side of the median. Repeat the argument for another median. So the centroid must be at the intersection of two medians. This is not a proof; it is, rather, a heuristic argument designed to produce belief. - Doctor Jerry, The Math Forum Check out our Web site http://mathforum.org/dr.math/ |
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