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2024 AMC 12A Question 20

AMC12 · 2024 AMC 12A

Question 20

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 $\triangle ABC?$
  1. $\left[\tfrac38,\tfrac12\right]$
  2. $\left(\tfrac12,\tfrac23\right]$
  3. $\left(\tfrac23,\tfrac34\right]$
  4. $\left(\tfrac34,\tfrac78\right]$
  5. $\left(\tfrac78,1\right]$