A point $P$ is randomly selected from the [rectangular](https://artofproblemsolving.com/wiki/index.php/Rectangle) region with vertices $(0,0),(2,0),(2,1),(0,1)$. What is the [probability](https://artofproblemsolving.com/wiki/index.php/Probability) that $P$ is closer to the [origin](https://artofproblemsolving.com/wiki/index.php/Origin) than it is to the point $(3,1)$?
$\mathrm{(A)}\ \frac 12 \qquad\mathrm{(B)}\ \frac 23 \qquad\mathrm{(C)}\ \frac 34 \qquad\mathrm{(D)}\ \frac 45 \qquad\mathrm{(E)}\ 1$
Solution 1
[2002 12B AMC-18.png](https://artofproblemsolving.com/wiki/index.php/File:2002_12B_AMC-18.png)

The region containing the points closer to $(0,0)$ than to $(3,1)$ is bounded by the [perpendicular bisector](https://artofproblemsolving.com/wiki/index.php/Perpendicular_bisector) of the segment with endpoints $(0,0),(3,1)$. The perpendicular bisector passes through midpoint of $(0,0),(3,1)$, which is $\left(\frac 32, \frac 12\right)$, the center of the [unit square](https://artofproblemsolving.com/wiki/index.php/Unit_square) with coordinates $(1,0),(2,0),(2,1),(1,1)$. Thus, it cuts the unit square into two equal halves of area $1/2$. The total area of the rectangle is $2$, so the area closer to the origin than to $(3,1)$ and in the rectangle is $2 - \frac 12 = \frac 32$. The probability is $\frac{3/2}{2} = \frac 34 \Rightarrow \mathrm{(C)}$.


Solution 2
Assume that the point $P$ is randomly chosen within the rectangle with vertices $(0,0)$, $(3,0)$, $(3,1)$, $(0,1)$. In this case, the region for $P$ to be closer to the origin than to point $(3,1)$ occupies exactly $\frac{1}{2}$ of the area of the rectangle, or $1.5$ square units.

If $P$ is chosen within the square with vertices $(2,0)$, $(3,0)$, $(3,1)$, $(2,1)$ which has area $1$ square unit, it is for sure closer to $(3,1)$.

Now if $P$ can only be chosen within the rectangle with vertices $(0,0)$, $(2,0)$, $(2,1)$, $(0,1)$, then the square region is removed and the area for $P$ to be closer to $(3,1)$ is then decreased by $1$ square unit, left with only $0.5$ square unit.

Thus the probability that $P$ is closer to $(3.1)$ is $\frac{0.5}{2}=\frac{1}{4}$ and that of $P$ is closer to the origin is $1-\frac{1}{4}=\frac{3}{4}$. $\mathrm{(C)}$

~ Nafer