Top piece above a quarter-height cut

Question

A point is chosen uniformly at random inside a triangle. A line parallel to the base is drawn one quarter 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 = 	frac14, the top triangle has height (1 - 	frac14) = 	frac34 of the whole, so its area fraction is P(	ext{top}) = \left(	frac34ight)^2 = \frac{9}{16} . Takeaway. A low cut leaves most of the triangle above it. The top area fraction is always (1 - h)^2 for a cut at height fraction h.

Top piece above a quarter-height cut

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