Ex 12.3, 5 - Find coordinates of points which trisect line segment

Ex 12.3,  5 - Chapter 12 Class 11 Introduction to Three Dimensional Geometry - Part 2
Ex 12.3,  5 - Chapter 12 Class 11 Introduction to Three Dimensional Geometry - Part 3 Ex 12.3,  5 - Chapter 12 Class 11 Introduction to Three Dimensional Geometry - Part 4

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Question 5 Find the coordinates of the points which trisect the line segment joining the points P (4, 2, –6) and Q (10, –16, 6). Let Point A (a, b, c) & point B (p, q, r) trisect the line segment PQ i.e. PA = AB = BC Point A divides PQ in the ratio of 1 : 2 We know that , Coordinate of point 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, m = 1 , n = 2 x1 = 4 , y1 = 2 , z1 = –6 x2 = 10 , y2 = –16 , z2 = 6 Coordinate of A are (a, b, c) = ((10 (1) + 4 (2))/(1 + 2),(−16 (1) + 2 (2))/(1 + 2),(6 (1) + (− 6) (2))/(1 + 2)) (a, b, c) = ((10 + 8)/3,(− 16 + 4)/3,(6 − 12)/3) (a, b, c) = (6, –4, –2) Hence, coordinates of A = (6, –4, –2) Now, Point B (p, q, r) divides AQ in the ratio 1 : 1 So, B is mid-point of AQ Coordinates of B = ((𝑥_(1 )+ 𝑥_2)/2,(𝑦_(1 )+ 𝑦_2)/2,(𝑧_(1 )+ 𝑧_2)/2) = ((6 + 10)/2,(−4 + (−16))/2,(−2 + 6)/2) = (160/2,(−20)/2,4/2) = (8, –10, 2) Hence coordinate of Point B = (8, –10, 2)

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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.