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Misc 1 - Three vertices of parallelogram ABCD are - Miscellaneous

Misc  1 - Chapter 12 Class 11 Introduction to Three Dimensional Geometry - Part 2
Misc  1 - Chapter 12 Class 11 Introduction to Three Dimensional Geometry - Part 3 Misc  1 - Chapter 12 Class 11 Introduction to Three Dimensional Geometry - Part 4

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Misc 1 Three vertices of a parallelogram ABCD are A (3, –1, 2), B (1, 2, –4) and C (–1, 1, 2). Find the coordinates of the fourth vertex. In parallelogram, diagonals bisect each other Hence AO = OC & BO = OD We can say that O is Midpoint of AC and O is Midpoint of BD Finding co-ordinates of O Now, O is the mid-point of AC Here, A (3, –1. 2 ) & C ( –1 , 1, 2) We know that If O (x, y, z) is the mid point of A (x1, y1, z1) & B (x2, y2, z2) , then coordinates of O O (x, y, z) = ((〖 𝑥〗_1 +〖 𝑥〗_2)/2,(〖 𝑦〗_1 + 〖 𝑦〗_2)/2,(𝑧_1 +〖 𝑧〗_2)/2) Here, x1 = 3, y1 = –1, z1 = 2 x2 = –1 , y2 = 1, z2 = 2 Putting values, O = ((3 + (−1))/2, (−1 + 1)/2, (2 + 2)/2) = ((3 − 1)/2, 0/2, 4/2) = (2/2, 0, 2) = (1, 0, 2) Thus, the co-ordinates of O is (1, 0, 2) Now, O is also the mid-point of BD , Finding coordinates of O, Here, B (1, 2, –4) , O (1, 0, 2) Let D Be (x, y, z) Now Co-ordinates of O = ((1 + 𝑥)/2,(2 + 𝑦)/2, (−4 + 𝑧)/2) (1, 0, 2) = ((1 + 𝑥)/2, (2 + 𝑦)/2, (−4 + 𝑧)/2) Thus, x = 1 , y = -2 , z = 8 ∴ Coordinates of point D are (1, –2, 8) Thus, Coordinates of the fourth vertex are (1, –2, 8) (1 + 𝑥)/2=1 1 + x = 2 x = 2 – 1 x = 1 (2 + 𝑦)/2=0 2 + y = 0 × 2 2 + y = 0 y = – 2 (−4 + 𝑧)/2=2 – 4 + z = 2 × 2 z = 4 + 4 z = 8

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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.