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Question:

If (1,2), (4,y), (x,6) and (3,5) are the vertices of a parallelogram taken in order, find x and y.

Solution:

Let A(1,2), B(4,y), C(x,6) and D(3,5) are the vertices of a parallelogram ABCD. AC and BD are the diagonals. O is the midpoint of AC and BD. If O is the mid-point of AC, then the coordinates of O are = ( (1+x)/2, (2+6)/2 ) = ( (x+1)/2, 4 ) If O is the mid-point of BD then coordinates of O are ( (4+3)/2, (5+y)/2 ) = ( 7/2, (5+y)/2 ) Since both coordinates are of the same point O ∴ (1+x)/2 = 7/2 ⇒ 1+x = 7 ⇒ x = 7-1 = 6 ∴ (5+y)/2 = 4 ⇒ 5+y = 8 ⇒ y = 8-5 = 3 Hence x = 6 and y = 3.