A random point above the midline of a triangle

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, splitting it into a small top triangle and a bottom trapezoid.

What is the probability the point lands in the small top triangle?

Accepted Answer

Cutting a triangle with a line parallel to the base, at height fraction h up from the base, leaves a top triangle that is similar to the whole. Its height is (1 - h) of the original, so every length scales by (1 - h) and its area scales by (1 - h)^2. Here the cut is at h = 	frac12, so the top triangle's height is 	frac12 of the whole and its area fraction is P(	ext{top}) = \left(1 - 	frac12ight)^2 = \left(	frac12ight)^2 = \frac14 . Takeaway. In a triangle, area scales as the square of height. A line at the midpoint of the height gives a top piece of only 	frac14 the area, not half, …

A random point above the midline of a triangle

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