AMC 12 2024a Problem 20

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

$\textbf{(A) } \left[\frac 38, \frac 12\right] \textbf{(B) } \left(\frac 12, \frac 23\right] \textbf{(C) } \left(\frac 23, \frac 34\right] \textbf{(D) } \left(\frac 34, \frac 78\right] \textbf{(E) } \left(\frac 78, 1\right]$