Question

If the three vertices of a parallelogram are (a + b, a - b), (2a + b, 2a - b) and (a - b, a + b) then the fourth vertex is

• A. (-a, a)
• B. (-a, -a)
• C. (-b, -b)
• D. None

Answer 1

The correct option is D None

Let the given vertices be $A(a+b,a-b),B(2a+b,2a-b)andC(a-b,a+b)$

Let the fourth vertex D $=(x,y)$

We know that the diagonals of a parallelogram bisect each other. So,the
midpoint of AC is same as the mid point of BD.
Mid point of two points ${(x}_1,y_1)$ and ${(x}_2,y_2)$ is calculated by the formula $(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$
So, midpoint of $AC=$ Mid point of $BD$
$=>(\frac{a+b+a-b}{2},\frac{a-b+a+b}{2})=(\frac{2a+b+x}{2},\frac{2a-b+y}{2})$
$=>(\frac{2a}{2},\frac{2a}{2})=(\frac{2a+b+x}{2},\frac{2a-b+y}{2})$
$=>2a+b+x=2a;2a-b+y=2a$
$=>x=-b;y=b$
Hence, $D=(-b,b)$