# What are the coordinates of S, if the midpoint of s (5, 4) and the coordinates of R are (9, 7)?

May 6, 2015

$\textcolor{red}{S} \textcolor{b l u e}{R}$ is a line segment with $\textcolor{b l u e}{R}$ at $\textcolor{b l u e}{\text{("9,7")}}$ and the midpoint of color(red(S)color(blue)(R) (let's call it $\textcolor{g r e e n}{T}$) at $\textcolor{g r e e n}{\text{("5,4")}}$

Let the coordinates of $\textcolor{red}{S}$ be $\textcolor{red}{\text{("x_s,y_s")}}$

Since $\textcolor{g r e e n}{T}$ is the midpoint of $\textcolor{red}{S} \textcolor{b l u e}{R}$

$\left(y \text{ at "color(blue)(R)) - (y " at "color(green)(T)) =(y " at " color(green)(T)) - (y " at } \textcolor{red}{S}\right)$
That is
$\textcolor{b l u e}{7} - \textcolor{g r e e n}{4} = \textcolor{g r e e n}{4} - \textcolor{red}{{y}_{s}}$
$\rightarrow \textcolor{red}{{y}_{s}} = 1$

Similarly for $\textcolor{red}{{x}_{s}}$
$\textcolor{b l u e}{9} - \textcolor{g r e e n}{5} = \textcolor{g r e e n}{5} - \textcolor{red}{{x}_{s}}$
$\rightarrow \textcolor{red}{{x}_{s}} = 1$

The coordinates of $\textcolor{red}{S}$ are $\textcolor{red}{\text{("1,1")}}$