Triangle Midpoints and VerticesDate: 02/23/2003 at 11:54:32 From: Katie Subject: Geometry How do you figure out the vertices of a triangle algebraically by using its three midpoints? Date: 02/23/2003 at 20:43:30 From: Doctor Greenie Subject: Re: Geometry Hi, Katie - This problem is solved very neatly using this fact: In any triangle, the line joining the midpoints of two sides is parallel to the third side and half the length of that third side. Then, given the midpoints A, B, and C of the three sides of a triangle, we know that (1) the two endpoints of the side of the triangle with midpoint A lie on a line containing A that is parallel to BC; the distance of each endpoint from point A is equal to the length of BC. (2) the two endpoints of the side of the triangle with midpoint B lie on a line containing B that is parallel to AC; the distance of each endpoint from point B is equal to the length of AC. (3) the two endpoints of the side of the triangle with midpoint C lie on a line containing C that is parallel to AB; the distance of each endpoint from point C is equal to the length of AB. A specific example shows that this method, while its description might sound somewhat complicated, is in fact very easy to use. Suppose the three given midpoints are A(-1,2), B(5,5), and C(3,-2). Let's find the endpoints of the side of the triangle with A as its midpoint, using (1) above. We need P and Q such that PA and QA are each parallel to BC and the same length as BC. Speaking informally, to get from B to C we have to move left 2 and down 7; so one of the endpoints of the side of the triangle with A as its midpoint is left 2 and down 7 from A(-1,2) - i.e., at (-3,-5). And to get from C to B we have to move right 2 and up 7; so the other endpoint of the side of the triangle with A as its midpoint is right 2 and up 7 from A(-1,2) - i.e., at (1,9). To find the endpoints of the side of the triangle with B as its midpoint, we see that to get from A to C we move right 4 and down 4; so to find the endpoints of the side of the triangle with B as its midpoint, we move right 4 and down 4 from B to get to (9,1), and we move left 4 and up 4 from B to get to (1,9). And to find the endpoints of the side of the triangle with C as its midpoint, we see that to get from A to B we move right 6 and up 3; so to find the endpoints of the side of the triangle with C as its midpoint, we move right 6 and up 3 from C to get to (9,1), and we move left 6 and down 3 from C to get to (-3,-5). So the three vertices of the original triangle, with midpoints A(-1,2), B(5,5), and C(3,-2), are (-3,-5), (1,9), and (9,1). Notice that we determined each of these vertices twice in our process; this gives us the opportunity to catch any arithmetic errors we might make. (The two answers we get for each vertex must be identical; if they are not, we made a mistake in our arithmetic.) For two algebraic approaches to this same problem (which are not only very different from the answer above but also very different from each other), take a look at the following pages in the Dr. Math archives: Finding Triangle Vertices from Midpoints http://mathforum.org/library/drmath/view/55193.html Using Midpoints to Determine Vertices http://mathforum.org/library/drmath/view/61165.html It is curious to note that one of these links shows an answer I myself provided to a similar question some time ago. Looking back at that answer (which uses an algebraic approach), it seems to me that the visual/geometric approach in the answer above is much easier to use. But you (or any other reader) might prefer an algebraic approach.... I hope all this helps. Please write back if you have any further questions about any of this. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/ |
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