Misc 3 - If origin is centroid of PQR with P (2a, 2, 6) - Section - Centroid

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Misc 3 If origin is the centroid of the triangle PQR with vertices P (2a, 2, 6), Q (–4, 3b, –10) and R (8, 14, 2c), then find the values of a, b and c. Given Δ PQR where P (2a, 2, 6) , Q (4, 3b, –10) , R (8, 14, 2c) Also, Origin O (0, 0, 0) is the centroid of Δ PQR We know that Co ordinate of centroid whose vertices are (x1 , y1 ,z1) ,(x2 ,y2 ,z2) (x3 ,y3 ,z3) is ﷐﷐﷐𝑥﷮1﷯ + ﷐𝑦﷮1﷯ + ﷐𝑧﷮1﷯﷮3﷯,﷐﷐𝑥﷮2﷯ + ﷐𝑦﷮2﷯ + ﷐𝑧﷮2﷯﷮3﷯,﷐﷐𝑥﷮3﷯ + ﷐𝑦﷮3﷯ + ﷐𝑧﷮3﷯﷮3﷯﷯ Here, x1 = 2a , y1 = 2 , z1 = 6 x2 = – 4 , y2 = 3b , z2 = –10 x3 = 8 , y2 = 14 , z3 = 2c ∴ Coordinates of centroid O(0, 0, 0) (0, 0, 0) = ﷐﷐2𝑎 + ﷐−4﷯ + 8﷮3﷯,﷐2 + 3𝑏 + 14﷮3﷯,﷐6 + ﷐−10﷯ + 2𝑐﷮3﷯﷯ (0, 0, 0) = ﷐﷐2𝑎 − 4 + 8﷮3﷯,﷐2 + 3𝑏 + 14﷮3﷯,﷐6 − 10 + 2𝑐﷮3﷯﷯ (0, 0, 0) = ﷐﷐2𝑎 + 4﷮3﷯,﷐3𝑏 + 16﷮3﷯,﷐2𝑐 − 4﷮3﷯﷯ Thus a = – 2 , b = ﷐−16﷮3﷯ & c= 2

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