# Difference between revisions of "2003 AMC 10A Problems/Problem 12"

## Problem

A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(4,0)$, $(4,1)$, and $(0,1)$. What is the probability that $x? $\mathrm{(A) \ } \frac{1}{8}\qquad \mathrm{(B) \ } \frac{1}{4}\qquad \mathrm{(C) \ } \frac{3}{8}\qquad \mathrm{(D) \ } \frac{1}{2}\qquad \mathrm{(E) \ } \frac{3}{4}$

## Solution

The rectangle has a width of $4$ and a height of $1$.

The area of this rectangle is $4\cdot1=4$.

The line $x=y$ intersects the rectangle at $(0,0)$ and $(1,1)$.

The area which $x>y$ is the right isosceles triangle with side length $1$ that has vertices at $(0,0)$, $(1,1)$, and $(0,1)$.

The area of this triangle is $\frac{1}{2}\cdot1^{2}=\frac{1}{2}$

Therefore, the probability that $x is $\frac{\frac{1}{2}}{4}=\frac{1}{8} \Rightarrow A$

## See Also

 2003 AMC 10A (Problems • Answer Key • Resources) Preceded byProblem 11 Followed byProblem 13 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 10 Problems and Solutions
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