Question

Three vertices of a rhombus taken in order are $(2,-1),(3,4)$ and $(-2,3)$ find the fourth vertex.

Answer 1

Let the vertices of a rhombus are $A(2,-1),B(3,4)$ and $C(-2,3).$

Let fourth vertex be $D(x,y).$

We know that diagonals of rhombus bisect each other,

$\therefore$ Co-ordinates of mid-points of $AC=$ Co-ordinates of mid-points of $BD$

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

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

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

$\Rightarrow$ $3+x=0$ and $4+y=2$

$\Rightarrow$ $x=-3$ and $y=-2$

$\therefore$ Co-ordinates of fourth vertex is $D(-3,-2).$