Question

The three vertices of a rhombus, taken in order are $(2,-1),(3,4)$ and $(-2,3)$. Then the fourth vertex is:

• A. $(3,-2)$
• B. $(-3,2)$
• C. $(3,2)$
• D. $(-3,-2)$

Answer 1

The correct option is D $(-3,-2)$

Let $A(2,-1),B(3,4),C(-2,3)$ and $D(x,y)$.

We know that, diagonals of rhombus bisect each other.

$\therefore$ Co-ordinates of mid-point of $AC=$ Co-ordinate of mid-point of $BD$.

$\Rightarrow$ $(\frac{2-2}{2},\frac{-1+3}{2})=(\frac{3+x}{2},\frac{4+y}{2})$

$\Rightarrow$ $(0,1)=(\frac{3+x}{2},\frac{4+y}{2})$

Now, we are going to co-ordinates,

$\Rightarrow$ $0=\frac{3+x}{2}$

$\Rightarrow$ $0=3+x$

$\therefore$ $x=-3$

$\Rightarrow$ $1=\frac{4+y}{2}$

$\Rightarrow$ $2=4+y$

$\Rightarrow$ $y=-2$

$\therefore$ The fourth vertex of rhombus is $D(-3,-2)$