19. A rectangle with vertices $(0,0),(100,0),(100,50)$ and $(0,50)$ has a circle with centre $(75,30)$ and radius 10 cut out of it. What is the slope of the line through the point $(75,30)$ which divides the remaining area of the rectangle into two shapes of equal area?
A $\frac{1}{5}$
B $\frac{1}{3}$
C $\frac{1}{2}$
D $\frac{2}{5}$
E $\frac{2}{3}$
SolUtion
A

Any line that bisects the area of the circle must pass through its centre. Similarly, any line that bisects the area of the rectangle must pass through the centre of the rectangle.
So, since the circle lies wholly within the rectangle, the line passing through the centre of the rectangle $(50,25)$ and the centre of the circle $(75,30)$ must cut the remaining area in half, as a semicircle is cut out of each side. The slope of that line is
$$
\frac{30-25}{75-50}=\frac{1}{5} \text {. }
$$