Medians of a trianglecoordinate free Assume that u, v, and

Chapter 11, Problem 81

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Medians of a trianglecoordinate free Assume that u, v, and w are vectors in _3 that form the sides of a triangle (see figure). Use the following steps to prove that the medians intersect at a point that divides each median in a 2:1 ratio. The proof does not use a coordinate system. M1 v u M3 M2 w O a. Show that u + v + w = b. Let M1 be the median vector from the midpoint of u to the opposite vertex. Define M2 and M3 similarly. Using the geometry of vector addition show that M1 = u>2 + v. Find analogous expressions for M2 and M3. c. Let a, b, and c be the vectors from O to the points one-third of the way along M1, M2, and M3, respectively. Show that a = b = c = 1u - w2>3. d. Conclude that the medians intersect at a point that divides each median in a 2:1 ratio. 8