**Xavier Décoret**

The distance between a point and a line is defined as the smallest distance between any point on the line and :

The manhattan distance between two points is defined as:

The question is then ``what is the formula that gives the manhattan distance between a point and a line?''.

Thus, if , we have and otherwise .

Next, we need to compute the two distances. For the horizontal one, we must solve:

which gives . The distance is then . Doing the same for the vertical one yields the . We conclude by noting that .