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Example 9 - Find coordinates of centroid of triangle - Examples

Example 9 - Chapter 12 Class 11 Introduction to Three Dimensional Geometry - Part 2
Example 9 - Chapter 12 Class 11 Introduction to Three Dimensional Geometry - Part 3

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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 โˆ† 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 ((๐‘ฅ_2 + ๐‘ฅ_3)/2,(๐‘ฆ_2 + ๐‘ฆ_3)/2,(๐‘ง_2 + ๐‘ง_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,) = ((ใ€–๐‘š ๐‘ฅใ€—_2 + ใ€– ๐‘› ๐‘ฅใ€—_1)/(๐‘š + ๐‘›),(ใ€–๐‘š ๐‘ฆใ€—_2+ใ€– ๐‘› ๐‘ฆใ€—_1)/(๐‘š + ๐‘›),(ใ€–๐‘š ๐‘งใ€—_2+ใ€– ๐‘› ๐‘งใ€—_1)/(๐‘š + ๐‘›)) Here, x1 = x1 , y1 = y1, z1 = z1 x2 = (๐‘ฅ_(2 )+ ๐‘ฅ_3)/2, y2 = (๐‘ฆ_2 +ใ€– ๐‘ฆใ€—_3)/2, z2 = (๐‘ง_(2 )+ ๐‘ง_3)/2 m = 2 , n = 1 Co-ordinate of G are = ((2((๐‘ฅ_(2 )+ ๐‘ฅ_3)/2) + 1 (๐‘ฅ1))/(2 + 1),(2 ((๐‘ฆ_2 +ใ€– ๐‘ฆใ€—_3)/2) + x1)/(2+1),(2 ((๐‘ง_(2 )+ ๐‘ง_3)/2) + z1)/(2 + 1)) = ((๐‘ฅ_2 + ๐‘ฅ_3 + ๐‘ฅ_1)/3,(๐‘ฆ_2 + ๐‘ฆ_3 + ๐‘ฆ_1)/3,(๐‘ง_2 + ๐‘ง_3 + ๐‘ง_1)/3) Hence Coordinate of centroid are ((๐’™_๐Ÿ + ๐’™_๐Ÿ + ๐’™_๐Ÿ‘)/๐Ÿ‘,(๐’š_๐Ÿ + ๐’š_๐Ÿ + ๐’š_๐Ÿ‘)/๐Ÿ‘,(๐’›_๐Ÿ + ๐’›_๐Ÿ + ๐’›_๐Ÿ‘)/๐Ÿ‘) Hence proved

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.