The area of the triangle formed by the axes and the line x+y= l corresponds to all the possibilities of breaking the stick into three parts i.e. x,y>0 & x+y<l.
Now, for the three pieces to form the sides of a triangle, the triangle equality should be true.
x+y>l-x-y ⇒ x+y> l/2.
x+l-x-y>y ⇒ y<l/2
y+l-x-y>x⇒ x<l/2
The shaded area in the figure above corresponds to all the three conditions above.
probability = favorable outcome / all possible outcomes
= shaded area/ total area
= (1/2 *l/2*l/2)/1/2 *l*l
= 1/4
