Question

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 $\triangle ABC?$

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

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

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

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

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

Solution:

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 $\int_{1/2}^{1}\left(1-\frac{1}{2x}\right)dx=\frac12-\frac{\ln2}{2}\approx0.153.$ 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 Problem 20

Solution:

Problem 20 in Other Years