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

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) = 1 + 2 2 , 1 + 2 2 , 1 + 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 + 2 , 2 + 2 , 4 + 2 (1, 0, 2) = 1 + 2 , 2 + 2 , 4 + 2 1 + 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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