Question

In the diagram, $PQRS$ is a straight line. $R$ is the midpoint of $QS$ and $Q$ is the midpoint of $PS$. $S$ is $(6,5)$ and $R$ is $(3,5)$. Find the coordinates of $P$.

• A. $(0,5)$
• B. $(-6,5)$
• C. $(-3,4)$
• D. $(-5,6)$

Answer 1

The correct option is B $(-6,5)$

According to the midpoint formula, the coordinates of the midpoint of the points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right).$

Let, the coordinates of the point $P$ and $Q$ are $(x,y)$ and $(a,b)$ respectively.

It is given that $R$ is the midpoint of $QS$ so, by using midpoint formula,

$(\frac{a+6}{2},\frac{b+5}{2})=(3,5)$

$\Rightarrow \frac{a+6}{2}=3$ and $\frac{b+5}{2}=5$

$\Rightarrow a+6=6$ and $b+5=10$

$\Rightarrow a=0$ and $b=5$

Hence, coordinate of the point $Q$ is $(0,5).$

Similarly, $Q$ is the midpoint of $PS$ so,

$(\frac{x+6}{2},\frac{y+5}{2})=(0,5)$

$\Rightarrow \frac{x+6}{2}=0$ and $\frac{y+5}{2}=5$

$\Rightarrow x+6=0$ and $y+5=10$

$\Rightarrow x=-6$ and $y=5$

Hence, coordinate of the point $P$ is $(-6,5).$