Examples

Chapter 11 Class 11 - Intro to Three Dimensional Geometry
Serial order wise

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Question 3 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 โ ABC Let AD be the median of ฮ ABC So, D is the mid point of BC Mid point of B(x2, y2, z2) and C(x3, y3, z3) is D ((๐ฅ_2 + ๐ฅ_3)/2,(๐ฆ_2 + ๐ฆ_3)/2,(๐ง_2 + ๐ง_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,) = ((ใ๐ ๐ฅใ_2 + ใ ๐ ๐ฅใ_1)/(๐ + ๐),(ใ๐ ๐ฆใ_2+ใ ๐ ๐ฆใ_1)/(๐ + ๐),(ใ๐ ๐งใ_2+ใ ๐ ๐งใ_1)/(๐ + ๐)) Here, x1 = x1 , y1 = y1, z1 = z1 x2 = (๐ฅ_(2 )+ ๐ฅ_3)/2, y2 = (๐ฆ_2 +ใ ๐ฆใ_3)/2, z2 = (๐ง_(2 )+ ๐ง_3)/2 m = 2 , n = 1 Co-ordinate of G are = ((2((๐ฅ_(2 )+ ๐ฅ_3)/2) + 1 (๐ฅ1))/(2 + 1),(2 ((๐ฆ_2 +ใ ๐ฆใ_3)/2) + x1)/(2+1),(2 ((๐ง_(2 )+ ๐ง_3)/2) + z1)/(2 + 1)) = ((๐ฅ_2 + ๐ฅ_3 + ๐ฅ_1)/3,(๐ฆ_2 + ๐ฆ_3 + ๐ฆ_1)/3,(๐ง_2 + ๐ง_3 + ๐ง_1)/3) Hence Coordinate of centroid are ((๐_๐ + ๐_๐ + ๐_๐)/๐,(๐_๐ + ๐_๐ + ๐_๐)/๐,(๐_๐ + ๐_๐ + ๐_๐)/๐) Hence proved
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