Top piece above a one-third cut

Question

A point is chosen uniformly at random inside a triangle. A line parallel to the base is drawn one third of the way up from the base.

What is the probability the point lands in the top piece (above the line)?

Accepted Answer

The cut is at height fraction h = 	frac13 up from the base, so the top triangle is similar to the whole with height (1 - 	frac13) = 	frac23 of the original. Its area fraction is the square of that: P(	ext{top}) = \left(	frac23ight)^2 = \frac{4}{9} . Takeaway. A cut lower down (closer to the base) leaves a bigger top triangle. At h = 	frac13, the top holds \left(	frac23ight)^2 = 	frac49 of the area.

Top piece above a one-third cut

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