first, we can subtract the area of the white from the area of the square. The top left triangle is x by 2x so the area is x^2. The triangle on the right is 3x by 4x so the area is 6x^2 the area of the bottom triangle is 2x by 4x so the area of that is 4x^2. the area of the whole square is 4x*4x=16x^2 16x^2-11x^2=5x^2
Suppose the points where we break the line are x and y, where 0 < x < y < 1. So our "possible" region is the following triangle on the plane:
The three sides have lengths x, y-x, and 1-y. We must satisfy the Triangle Inequality on all three sides, meaning that the sum of any two of these lengths must be greater than the third. This gives us
1. : This gives y>1-y, so y> 1/2.
2. : This gives 1-x> x, so x<1/2.
3. : This gives , so .
Adding these lines to the diagram gives us the darker "success" region.
The darker region makes up 1/4 of the total region, so the probability is 1/4.