A random chord longer than the side of the triangle

Question

A circle has an equilateral triangle inscribed in it. A chord of the circle is chosen "at random".

What is the probability that the chord is longer than a side of the triangle?

Try three natural ways of choosing a random chord: (a) pick its two endpoints at random on the circle; (b) pick a random radius, then a random point on that radius, and draw the chord through that point perpendicular to the radius; (c) pick a random point inside the circle and draw the chord that has that point as its midpoint. Show that they give three different answers, and explain what that means.

Accepted Answer

Method (a) gives 1/3, method (b) gives 1/2, method (c) gives 1/4. The question is not well posed until you say what "random chord" means. The geometry. The side of the inscribed equilateral triangle sits at distance R/2 from the centre (R the radius). A chord is longer than that side exactly when it is closer to the centre than R/2, that is, when its midpoint lies within the concentric circle of radius R/2. (a) Random endpoints. Fix one endpoint at a vertex of the triangle (by symmetry this loses nothing). The chord is long exactly when the other endpoint lies on the arc between the other two…

A random chord longer than the side of the triangle

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