Question

在等边三角形 $\triangle ABC$ 中，点 $P$ 和 $Q$ 分别在边 $\overline{AB}$ 与 $\overline{AC}$ 上独立均匀随机选取。下面哪个区间包含 $\triangle APQ$ 的面积小于 $\triangle ABC$ 面积一半的概率？

Points $P$ and $Q$ are chosen uniformly and independently at random on sides $\overline{AB}$ and $\overline{AC},$ respectively, of equilateral triangle $\triangle ABC.$ Which of the following intervals contains the probability that the area of $\triangle APQ$ is less than half the area of $\triangle ABC?$

$\left[\tfrac38,\tfrac12\right]$

$\left(\tfrac12,\tfrac23\right]$

$\left(\tfrac23,\tfrac34\right]$

$\left(\tfrac34,\tfrac78\right]$

$\left(\tfrac78,1\right]$

答案：D

知识点：几何概率面积比

难度评级：2100

解答：

令 $x=\tfrac{AP}{AB}$ ， $y=\tfrac{AQ}{AC}$ ，二者在 $[0,1]$ 上均匀分布，则面积比 $\tfrac{[APQ]}{[ABC]}=xy$ 。补事件 $xy\ge\tfrac12$ 要求 $x\ge\tfrac12$ ，且 $y\in[\tfrac{1}{2x},1]$ ，其概率为因此 $P(xy\lt\tfrac12)\approx1-0.153=0.847$ ，落在 $\left(\tfrac34,\tfrac78\right]$ 中。因此正确答案是 D。 $\begin{aligned} &\int_{1/2}^{1}\left(1-\frac{1}{2x}\right)dx \\ &=\frac12-\frac{\ln2}{2}\approx0.153. \end{aligned}$

With $x=\tfrac{AP}{AB}$ and $y=\tfrac{AQ}{AC}$ uniform on $[0,1],$ the area ratio $\tfrac{[APQ]}{[ABC]}=xy.$ The complementary event $xy\ge\tfrac12$ requires $x\ge\tfrac12$ and $y\in[\tfrac{1}{2x},1],$ with probability $\begin{aligned} &\int_{1/2}^{1}\left(1-\frac{1}{2x}\right)dx \\ &=\frac12-\frac{\ln2}{2}\approx0.153. \end{aligned}$ Therefore $P(xy\lt\tfrac12)\approx1-0.153=0.847,$ which lies in $\left(\tfrac34,\tfrac78\right].$ Thus, the correct answer is D.

2024 AMC 12A 第 20 题

答案：D

难度评级：2100

解答：

其他年份的第 20 题