AMC10 2003 A

AMC10 2003 A · Q12

AMC10 2003 A · Q12. It mainly tests Area & perimeter, Geometric probability (basic).

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 $x < y$?

从矩形内部随机选取一点 $(x, y)$，该矩形的顶点为 $(0, 0)$、$(4, 0)$、$(4, 1)$ 和 $(0, 1)$。$x < y$ 的概率是多少？

(A) $\frac{1}{8}$ $\frac{1}{8}$

(B) $\frac{1}{4}$ $\frac{1}{4}$

(D) $\frac{1}{2}$ $\frac{1}{2}$

(E) $\frac{3}{4}$ $\frac{3}{4}$

Answer

Correct choice: (A)

正确答案：（A）

Solution

(A) The point $(x,y)$ satisfies $x<y$ if and only if it belongs to the shaded triangle bounded by the lines $x=y$, $y=1$, and $x=0$, the area of which is $1/2$. The ratio of the area of the triangle to the area of the rectangle is $\frac{1/2}{4}=\frac{1}{8}$.

（A）点$(x,y)$满足$x<y$当且仅当它位于由直线$x=y$、$y=1$和$x=0$围成的阴影三角形内，该三角形的面积为$1/2$。三角形面积与矩形面积之比为$\frac{1/2}{4}=\frac{1}{8}$。

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