If three consecutive vertices of a parallelogram are (1, -2), (3, 6) and (5, 10), find its fourth vertex.
Let three vertices be A(1, -2), B(3, 6) and C(5, 10) and fourth vertex be D(x, y)
It is given that quadrilateral joining these four vertices is parallelogram, ie □ABCD is parallelogram.
We know that diagonals of parallelogram bisect each other, ie midpoint of the diagonals coincide.
Let E(xm , ym) be the midpoint of diagonals AC and BD.
By midpoint formula,
x = , y =
For diagonal AC,
xm = , ym =
∴ xm = , ym =
∴ E(xm , ym) ≡ (3, 4)
For diagonal BD,
3 = , 4 =
∴ x = 6 – 3 , y = 8 – 6
∴ x = 3 and y = 2
Hence, our fourth vertex is D(3 , 2)