AMC 12A 2024 (Problem 20)

ProbabilityHard

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

\text{(A)}\;\left[\frac{3}{8},\frac{1}{2}\right]

\text{(B)}\;\left(\frac{1}{2},\frac{2}{3}\right]

\text{(C)}\;\left(\frac{2}{3},\frac{3}{4}\right]

\text{(D)}\;\left(\frac{3}{4},\frac{7}{8}\right]

\text{(E)}\;\left(\frac{7}{8},1\right]