## Friday, 9 November 2012

### A Circle, A Triangle and Some Random Lines

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Say you have a circle in which there is an equilateral triangle, like this.

If you pick, at random, a line which passes through the circle, what is the probability that the section of your line that lies within the circle will be longer than the sides of the equilateral triangle?

The three red segments that make up the triangle are, of course, the same length as the sides of the triangle.
If you need some more visual input to work out what I mean, here are three random lines you could choose:

For those who are frightened of probability and statistics, you might want to try the simpler question:  are there more shorter lines, more longer lines or most equally long lines?

It's worth trying to figure it out yourself before looking at the answer - here.

1. I believe the answer is 1/3. In order to be longer than the side of the triangle, the line if it started intersecting the circle at any randomly chosen point, would need to intersect the circle again at a point on the circle greater than 120 degrees around the circle, but less than 240 degrees. That means 1/3 of the circle is open for that line to intersect with.

1. Thanks Daniel. You've clearly thought about it.

How many agree with this answer?

2. That was the answer I got as well. I just considered one point anchored to where the triangle and circle share a point (not necessarily true, but we could rotate the triangle to make it so without altering any lengths) and swept out the regions to compare the line segment lengths. 2/3 of all possible lines fall shorter than the triangle side length, and 1/3 come out to be longer.

3. I'd not be surprised if that is the result that most people come up with. It's not right, but it certainly seems right at first blush.

2. 1/2. each angle is 60 degrees of lines longer than the side of the equilateral triangle, for a total of 180 degrees, half the 360 degree circumference of the circle. there are an equal number of short and long lines.

1. My calculations, and the exhaustive argumentation I provide in follow up articles, arrive at p=0.5 as the correct answer. I don't quite understand what you mean here though. Could you possibly draw what you mean? (Sorry, I am a hopelessly visual person.)

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