In the diagram, point $Q$ is the midpoint of $P R$. The coordinates of $R$ are
(A) $(2,5)$
(B) $(7,11)$
(C) $(6,9)$
(D) $(8,10)$
(E) $(9,15)$

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-044.jpg?height=396&width=464&top_left_y=1775&top_left_x=1210)
Solution 1

Let the coordinates of $R$ be $(a, b)$.

Since $Q$ is the midpoint of $P$ and $R$, then the difference between the $x$-coordinates of $Q$ and $P$ equals the difference between the $x$-coordinates of $R$ and $Q$.

In other words, $a-4=4-1$ and so $a=7$.

Similarly, $b-7=7-3$ and so $b=11$.

Thus, the coordinates of $R$ are $(7,11)$.

## Solution 2

Let the coordinates of $R$ be $(a, b)$.

The midpoint of $P(1,3)$ and $R(a, b)$ has coordinates $\left(\frac{1}{2}(1+a), \frac{1}{2}(3+b)\right)$.

Since $Q(4,7)$ is the midpoint of $P R$, then $4=\frac{1}{2}(1+a)$ (which gives $7=1+a$ or $a=7$ ) and $14=\frac{1}{2}(3+b)($ which gives $14=3+b$ or $b=11)$.

Therefore, the coordinates of $R$ are $(7,11)$.

ANSWER: (B)