Problem 115

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Problem 115: (2024 AMC 12a, 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 $\triangle ABC?$ $\textbf{(A) } \left[\frac 38, \frac 12\right] \qquad$ $ \textbf{(B) } \left(\frac 12, \frac 23\right] \qquad$ $ \textbf{(C) } \left(\frac 23, \frac 34\right] \qquad$ $ \textbf{(D) } \left(\frac 34, \frac 78\right] \qquad$ $ \textbf{(E) } \left(\frac 78, 1\right]$

Solution

Exercise 115: Solution

Let $ABC$ be a triangle. Points $P$ and $Q$ are chosen uniformly and independently at random on sides $\overline{AB}$ and $\overline{AC}$, respectively. What is the probability that the area of $\triangle APQ$ is less than one third of the area of $\triangle ABC$?

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