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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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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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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.