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Perpendiculars Picking Probability (Posted on 2005-11-24) Difficulty: 3 of 5
You choose a random point, uniformly, within an equilateral triangle.

What's the average distance to the three sides?

See The Solution Submitted by Old Original Oskar!    
Rating: 3.5000 (4 votes)

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Some Thoughts Quick and dirty solution; not a proof Comment 6 of 6 |
If the answer is the same for any point within the triangle, then I can pick any point I want and solve for that point.
I choose one of the vertices of the triangle.
The distance to 2 of the sides is zero.
The distance to the other side is just the altitude of an equilateral triangle which is sqrt(3)/2.

average(0,0,sqrt(3)/2) = sqrt(3)/6

I haven't proved that the answer is the same for every point in the triangle, but the statement of the problem implies that I can assume it.  So:  not a rigorous solution, but if this were a question on a standardized test, you could save a lot of time, get the right answer, and move on to the next question.

  Posted by Larry on 2020-04-23 06:48:48
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