Top piece above a two-thirds cut

Question

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

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

Accepted Answer

With the cut at height fraction h = 	frac23, the top triangle has height (1 - 	frac23) = 	frac13 of the whole, so its area fraction is P(	ext{top}) = \left(	frac13ight)^2 = \frac{1}{9} . Takeaway. Near the apex the triangle is narrow, so a high cut isolates only a tiny sliver: at h = 	frac23 the top is 	frac19 of the area.

Top piece above a two-thirds cut

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