Example 9 - Find coordinates of centroid of triangle - Section - Centroid

  1. Chapter 12 Class 11 Introduction to Three Dimensional Geometry
  2. Serial order wise
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Example 9 Find the coordinates of the centroid of the triangle whose vertices are (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3). Let ABC be the triangle where A (x1, y1, z1), B(x2, y2, z2) and C(x3, y3, z3), We need to find co- ordinate of centroid. Let G be the centroid of ∆ ABC Let AD be the median of Δ ABC So, D is the mid point of BC Mid point of B(x2, y2, z2) and C(x3, y3, z3) is D ﷐﷐﷐𝑥﷮2﷯ + ﷐𝑥﷮3﷯﷮2﷯,﷐﷐𝑦﷮2﷯ + ﷐𝑦﷮3﷯﷮2﷯,﷐﷐𝑧﷮2﷯ + ﷐𝑧﷮3﷯﷮2﷯﷯ We know that centroid divides median in the 2 : 1 So, centroid (G) divides the median AD in the ratio of 2 : 1 We know that Co-ordinate of point P (x, y, z) that divides the line segment joining A (x1, y1, z1) & B (x2, y2, z2) internally in the ratio m : n is P (x, y, z,) = ﷐﷐﷐𝑚 𝑥﷮2﷯+﷐ 𝑛 𝑥﷮1﷯﷮𝑚+𝑛﷯,﷐﷐𝑚 𝑦﷮2﷯+﷐ 𝑛 𝑦﷮1﷯﷮𝑚+𝑛﷯,﷐﷐𝑚 𝑧﷮2﷯+﷐ 𝑛 𝑧﷮1﷯﷮𝑚+𝑛﷯﷯ Here, x1 = x1 , y1 = y1, z1 = z1 x2 = ﷐﷐𝑥﷮2 ﷯+ ﷐𝑥﷮3﷯﷮2﷯, y2 = ﷐﷐𝑦﷮2﷯ +﷐ 𝑦﷮3﷯﷮2﷯, z2 = ﷐﷐𝑧﷮2 ﷯+ ﷐𝑧﷮3﷯﷮2﷯ m = 2 , n = 1 Co-ordinate of G are = ﷐﷐2﷐﷐﷐𝑥﷮2 ﷯+ ﷐𝑥﷮3﷯﷮2﷯﷯ + 1 (𝑥1)﷮2 + 1﷯,﷐2 ﷐﷐﷐𝑦﷮2﷯ +﷐ 𝑦﷮3﷯﷮2﷯﷯ + x1﷮2+1﷯,﷐2 ﷐﷐﷐𝑧﷮2 ﷯+ ﷐𝑧﷮3﷯﷮2﷯﷯ + z1﷮2 + 1﷯﷯ = ﷐﷐﷐𝑥﷮2﷯ + ﷐𝑥﷮3﷯ + ﷐𝑥﷮1﷯﷮3﷯,﷐﷐𝑦﷮2﷯ + ﷐𝑦﷮3﷯ + ﷐𝑦﷮1﷯﷮3﷯,﷐﷐𝑧﷮2﷯ + ﷐𝑧﷮3﷯ + ﷐𝑧﷮1﷯﷮3﷯﷯ Hence Coordinate of centroid are ﷐﷐﷐𝑥﷮1﷯ + ﷐𝑥﷮2﷯ + ﷐𝑥﷮3﷯﷮3﷯,﷐﷐𝑦﷮1﷯ + ﷐𝑦﷮2﷯ + ﷐𝑦﷮3﷯﷮3﷯,﷐﷐𝑧﷮1﷯ + ﷐𝑧﷮2﷯ + ﷐𝑧﷮3﷯﷮3﷯﷯ Hence proved

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