A line segment has one endpoint at $(3,9)$ and midpoint at $(1,2)$. What is the sum of the coordinates of the other endpoint?
To find the other endpoint, I need to use the fact that the midpoint is the average of the endpoints. That means that the midpoint's x-coordinate is the average of the endpoints' x-coordinates, and the same for the y-coordinates. Let me call the other endpoint $(x,y)$. Then I have the equations $1 = \frac{3+x}{2}$ and $2 = \frac{9+y}{2}$. Solving for $x$ in the first equation, I get $x = 2\cdot 1 - 3 = -1$. Solving for $y$ in the second equation, I get $y = 2\cdot 2 - 9 = -5$. So the other endpoint is $(-1,-5)$. The sum of the coordinates of the other endpoint is $-1 + (-5) = -6$. 