Bottom trapezoid below the midline

Question

A point is chosen uniformly at random inside a triangle. A line parallel to the base is drawn at half the triangle's height.

What is the probability the point lands in the bottom trapezoid (below the line)?

Accepted Answer

The top triangle above a half-height cut is similar to the whole with height 	frac12, so its area fraction is \left(	frac12ight)^2 = 	frac14. The bottom trapezoid is everything else: P(	ext{bottom}) = 1 - \frac14 = \frac34 . Takeaway. The wide part of a triangle is near the base, so the lower half by height carries the bulk of the area, here 	frac34.

Bottom trapezoid below the midline

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