Question

A line is divided at random into three parts. What is the probability that they form the sides of a triangle?

Answer 1

Let AP = x, BQ = y and AB = a. Then length of third side is a – x – y.

Since the sum of two sides of a triangle is greater than the third side, AP must be less than a/2, that is, x < a/2. Similarly, y < a/2 and

a - (x + y) < a/2 or x + y > a/2   (1)

For all possible cases of dividing the line

Condition (2) corresponds to the triangular region OXY and 

condition (1) corresponds to the triangular region PQR (Fig.).