2002 AMC 12B Problem 18

The points closer to $(0,0)$ than to $(3,1)$ lie on the origin side of the perpendicular bisector of that segment, the line $3x+y=5.$ Within the rectangle, this region is a trapezoid whose parallel sides have lengths $\dfrac53$ (at $y=0$) and $\dfrac43$ (at $y=1$), so its area is $\dfrac12\left(\dfrac53+\dfrac43\right)=\dfrac32.$ The rectangle has area $2,$ so the probability is $\dfrac{\frac{3}{2}}{2}=\dfrac34.$ Thus, the correct answer is C.