2003 AMC 10A Problem 12

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 \lt y?

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Solution

The condition

x \lt y

holds in the triangle bounded by

y = x,

y = 1,

and

x = 0,

which has vertices

(0, 0),

(0, 1),

and

(1, 1).

This triangle has area

\dfrac{1}{2},

while the rectangle has area

4.

The probability is

\dfrac{\frac{1}{2}}{4} = \dfrac{1}{8}.

Thus, the correct answer is A.