The three vertices of a rhombus PQRS are $\displaystyle \mathrm{P}(2,-3), \mathrm{Q}(6,5)$ and $\displaystyle \mathrm{R}(-2,1)$. Find the coordinates of the fourth vertex S and coordinates of the point where both the diagonals PR and QS intersect.

Let the coordinates of fourth vertex S be (x,y) Coordinates of mid-point of PR = Coordinates of mid-point of QS \(\displaystyle \left(\frac{\mathrm{x}+6}{2}, \frac{\mathrm{y}+5}{2}\right)=\left(\frac{2-2}{2}, \frac{-3+1}{2}\right)\) \(\displaystyle \mathrm{x}=-6\) \(\displaystyle \mathrm{y}=-7\) ∴ coordinates of \(\displaystyle \mathrm{S}=(-6,-7)\) mid- point of diagonal \(\displaystyle \mathrm{PR}=(0,-1)\)