Page "Bertrand paradox (probability)" Paragraph 4

# Random chords, selection method 2 The " random radius " method: Choose a radius of the circle, choose a point on the radius and construct the chord through this point and perpendicular to the radius.

To calculate the probability in question imagine the triangle rotated so a side is perpendicular to the radius.

The chord is longer than a side of the triangle if the chosen point is nearer the center of the circle than the point where the side of the triangle intersects the radius.

The side of the triangle bisects the radius, therefore the probability a random chord is longer than a side of the inscribed triangle is 1 / 2.