Question

If $(1,2),(3,4)$ and $(0,6)$ are the three vertices of a parallelogram taken in that order, then the fourth vertex is .......... .

• A. $(2,4)$
• B. $(-2,4)$
• C. $(4,2)$
• D. $(-4,2)$

Answer 1

Answer: (B)

$(-2,4)$

Let the vertices of the parallelogram be $A(1,2)$, $B(3,4)$, $C(0,6)$ and $D(x,y)$.

Thus, the diagonals are AC and BD.

Now diagonals of a parallelogram bisect each other.

Therefore, the midpoint of AC is the midpoint of BD.

Now the midpoint of AC is $(\frac{1+0}{2},\frac{2+6}{2})=(\frac{1}{2},4)$.

Now the midpoint of BD will be $(\frac{3+x}{2},\frac{4+y}{2})=(\frac{1}{2},4)$

$\frac{3+x}{2}=\frac{1}{2}$ or $x=-2$.

Similarly $\frac{4+y}{2}=4$ or $y=4$

Hence the last vertex is $(-2,4)$.