Question 1. Coordinates geometry ( Mid point concept)

1. If the vertices of $a$ parallelogram $P Q R S$ taken in order are $\mathrm{P}(3,4), \mathrm{Q}(-2,3)$ and $R(-3,-2)$, then the coordinates of its fourth vertex S are:
(a) $(-2,-1)$
(b) (-2, -3)
(c) $(2,-1)$
(d) $(1,2)$

Answer: (c) $(2,-1)$

Explanation: Since, the diagonals of $a$ parallelogram bisect each other.
Let the coordinates of $S$ be $(x, y)$.
∴ Coordinates of the mid point of $\mathrm{PR}=$ coordinates of the mid point of QS
$$
\begin{array}{llrl}
\Rightarrow & \left(\frac{3+(-3)}{2}, \frac{4+(-2)}{2}\right) & =\left(\frac{-2+x}{2}, \frac{3+y}{2}\right) \\
\Rightarrow & & (0,1) & =\left(\frac{x-2}{2}, \frac{3+y}{2}\right) \\
\Rightarrow & & \frac{x-2}{2} & =0 \text { and } \frac{3+y}{2}=1 \\
\Rightarrow & & x-2 & =0 \text { and } 3+y=2 \\
\Rightarrow & & x & =2 \text { and } y=-1
\end{array}
$$

Hence, the fourth vertex of the parallelogram is $(2,-1)$.