# Example 9 - Chapter 12 Class 11 Introduction to Three Dimensional Geometry

Last updated at May 29, 2018 by Teachoo

Last updated at May 29, 2018 by Teachoo

Transcript

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 ABC So, D is the mid point of BC Mid point of B(x2, y2, z2) and C(x3, y3, z3) is D 2 + 3 2 , 2 + 3 2 , 2 + 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,) = 2 + 1 + , 2 + 1 + , 2 + 1 + Here, x1 = x1 , y1 = y1, z1 = z1 x2 = 2 + 3 2 , y2 = 2 + 3 2 , z2 = 2 + 3 2 m = 2 , n = 1 Co-ordinate of G are = 2 2 + 3 2 + 1 ( 1) 2 + 1 , 2 2 + 3 2 + x1 2+1 , 2 2 + 3 2 + z1 2 + 1 = 2 + 3 + 1 3 , 2 + 3 + 1 3 , 2 + 3 + 1 3 Hence Coordinate of centroid are 1 + 2 + 3 3 , 1 + 2 + 3 3 , 1 + 2 + 3 3 Hence proved

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