Difference between revisions of "2024 AMC 12A Problems/Problem 20"
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Revision as of 17:49, 8 November 2024
Points and are chosen uniformly and independently at random on sides and respectively, of equilateral triangle Which of the following intervals contains the probability that the area of is less than half the area of
Solution 1
Let and . Applying the sine formula for a triangle's area, we see that
Without loss of generality, we let , and thus ; we therefore require for . A quick rough sketch of on the square given by reveals that the curve intersects the boundaries at and , and it is actually quite (very) obvious that the area bounded by the inequality and the aforementioned unit square is more than but less than (cf. the diagram below). Thus, our answer is .
~Technodoggo