Question

A line is divided at random into three parts, the probability that they form the sides of a possible triangle is

• A. $\frac{1}{2}$
• B. $\frac{2}{3}$
• C. $\frac{3}{4}$
• D. $\frac{1}{4}$

Answer 1

The correct option is D $\frac{1}{4}$

Consider an equilateral triangle with altitude $1$. It is not hard to show that if you choose a point randomly in this triangle, the distances to the three sides gives the same distribution of lengths that you obtain by breaking a stick at two random points. Now, the locus of points for which no distance is longer than $\frac{1}{2}$ is the smaller equilateral triangle formed by joining the midpoints of the edges, which has area $\frac{1}{2}$ that of the original triangle.