# Misc 1

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

Transcript

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) = 𝑥1 + 𝑥22, 𝑦1 + 𝑦22,𝑧1 + 𝑧22 Here, x1 = 3, y1 = – 1, z1 = 2 x2 = –1 , y2 = 1, z2 = 2 Putting values, O = 3+ (−1)2, −1+12, 2+22 = 3−12, 02, 42 = 22, 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 + 𝑥2,2 + 𝑦2, −4 + 𝑧2 (1, 0, 2) = 1 + 𝑥2, 2 + 𝑦2, −4 + 𝑧2 1 + 𝑥2=1 1 + x = 2 x = 2 – 1 x = 1 Thus, x = 1 , y = -2 , z = 8 ∴ Coordinates of point D are (1, – 2, 8) ⇒ Coordinates of the fourth vertex are (1, – 2, 8)

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.