Three vertices of a parallelogram are (a + b, a - b), (2a + b, 2a - b), (a - b, a + b). Find the fourth vertex.
Let A(a + b, a - b), B(2a + b, 2a - b), C(a - b, a + b) and fourth vertex be D(x, y).
It is given that □ABCD is parallelogram.
We know that diagonals of parallelogram bisect each other.
Let intersection of diagonals be E(xm, ym )
By midpoint formula.
xm = , ym =
For midpoint E of diagonal AC,
xm = , ym =
∴ xm = a , ym = a
∴E(xm, ym ) ≡ (a, a)
For diagonal BD,
a = , a=
∴ 2a = 2a + b +x , 2a = 2a – b +y
∴ x = -b and y = b
Hence, the fourth vertex is D(-b, b)