How Close

The coordinates of two points on a line are given. The task is to find the shortest distance from the line of a third given point. The task involves finding the equation of the line joining the first two points, and then calculating a normal to this line that passes through the third point. An alternative method of working is suggested using a circle centred on the third point with the given line as a tangent. This gives rise to solving a simultaneous equation with one of the equations non-linear.

The problem set contains three different lines and for each example the shortest distance is 21. The question then arises as to how it was possible to get integer value coordinates for all of the tasks. The investigation begins by looking at a unit circle and establishing points that have rational coordinates. The results may then be scaled up to give the different examples.