Medians of a trianglewith coordinates In contrast to the

Chapter 11, Problem 82

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Medians of a trianglewith coordinates In contrast to the proof in Exercise 81, we now use coordinates and position vectors to prove the same result. Without loss of generality, let P1x1, y1, 02 and Q1x2, y2, 02 be two points in the xy-plane and let R1x3, y3, z32 be a third point, such that P, Q, and R do not lie on a line. Consider _PQR. a. Let M1 be the midpoint of the side PQ. Find the coordinates of M1 and the components of the vector RM r 1. b. Find the vector OrZ 1 from the origin to the point Z1 two-thirds of the way along RM r 1. c. Repeat the calculation of part (b) with the midpoint M2 of RQ and the vector PM r 2 to obtain the vector OZ r 2. d. Repeat the calculation of part (b) with the midpoint M3 of PR and the vector QrM 3 to obtain the vector OZ r 3. e. Conclude that the medians of _PQR intersect at a point. Give the coordinates of the point. f. With P12, 4, 02, Q14, 1, 02, and R16, 3, 42, find the point at which the medians of _PQR intersect. 8