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Misc 1 - Three vertices of parallelogram ABCD are - Distance between two points - Verifying

  1. Chapter 12 Class 11 Introduction to Three Dimensional Geometry
  2. Serial order wise
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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﷯﷯ ﷐1 + 𝑥﷮2﷯=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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