Question

The three vertices of a parallelogram are (1, 1), (4, 4), (4, 8). find the fourth vertex

Answer 1

let ABCD be the parallelogram
and D be the fourth vertex of parallelogram
since the diagonals of parallelogram bisect each other
therefore Midpoint of AC= Midpoint of BD
By M.P. formula
(1+4/2,8+1/2)=(4+x/2,4+y/2)
(5/2,9/2)=(4+x/2,4+y/2)
on comparing;
x=1
y=5
therefore vertex D=(1,5)

Answer 2

Solution :

Let A(1, 1), B(4, 4), C(4, 8) and D(x, y) are the vertices of an triangle.

We know that diagonals of a parallelogram bisect each other

So, the coordinate of the mid-point of AC = the coordinate of the mid-point of BD

=> {(1 + 4)/2, (1 + 4)/2} = {(4 + x)/2, (8 + y)/2}

=> (5/2, 5/2) = {(4 + x)/2, (8 + y)/2}

=> (4 + x)/2 = 5/2 and (8 + y)/2 = 5/2

=> 4 + x = 5 and 8 + y = 5

=> x = 5 - 4 and y = 5 - 8

=> x = 1 and y = -3

So, the fourth vertex is (1, -3).

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