Vector Proof: Parallelogram Diagonals
Date: 01/20/99 at 14:27:50 From: Stephanie Subject: Vector proofs Dear Dr. Math, I have tried for a very long time to solve these proofs but I just don't get them. 1. Given: Parallelogram ABCD Prove: the diagonals of a parallelogram bisect each other (Note: I must use the method: Let X be the midpoint of vector AC and show that vector BX = vector XD) 2. Prove that the line joining the midpoints of two sides of a triangle is parallel to the third side and half its length. Thanks, Stephanie
Date: 01/20/99 at 20:32:19 From: Doctor Schwa Subject: Re: Vector proofs Question 1: Let ABCD be the vertices of the parallelogram in that order A ------- B \ \ \ \ D ------- C Then what you know is that the vector AB = DC and the vector AD = BC. You also know things like AB + BC = AC as vectors. Let's let O be the origin somewhere. Then the midpoint of AC is (OA + OC)/2 = OX. What's vector BX then? It's OX - OB. And XD = OD - OX. By combining the value I found for point X with some of these other things (and other relations, like AB = OB - OA = DC = OC - OD ...) you should be able to put together a proof. Question 2: If the triangle is ABC, then the midpoint of AB is found by the vector OM = (OA + OB)/2 and the midpoint of AC is OL = (OA + OC)/2 Then the vector from M to L is ML = OL - OM. See what you can do from there. The key point to recognize in both these proofs is that if you're going to have a vector, you need a starting and an ending point. It's convenient to have everything start from the same point. So instead of "vector A," which doesn't really make sense, say "vector OA." Then vectors like AB can be rewritten as OB - OA. When you've referred everything to a common standard (the origin) it's much easier to combine them. - Doctor Schwa, The Math Forum http://mathforum.org/dr.math/
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