Three vertices of a parallelogram are (a+b, a-b) . . .

Question : Three vertices of a parallelogram are (a+b, a-b), (2a+b, 2a-b), (a-b, a+b). Find the fourth vertex?

Doubt by Vanshika

Solution :

Let
A (a+b, a-b)

B (2a+b, 2a-b)

C (a-b, a+b)
D (x₄, y₄)

We know diagonals of parallelogram bisect each other. So
Mid Point of AC = Mid Point of BD
[(x₁+x₃)/2 , (y₁+y₃)/2] = [(x₂+x₄)/2 , (y₂+y₄)/2]
[(a+b+a-b)/2 , (a-b+a+b)/2] = [(2a+b+x₄)/2 , (2a-b+y₄)/2]

[(2a)/2 , (2a)/2] = [(2a+b+x₄)/2 , (2a-b+y₄)/2]

Equating the x and y coordinates both sides.

2a/2 = (2a+b+x₄)/2
2a=2a+b+x₄
2a-2a-b=x₄
x₄=-b

2a/2=(2a-b+y₄)/2
2a=2a-b+y₄
2a-2a+b=y₄

y₄=b

Hence, the coordinates of fourth vertex of the parallelogram will be (x₄,y₄) = (-b,b).