Take a paper and draw the following:

Let the centre of the circle be O. Let PQ be a tangent touching the circle at T. Clearly OT is a radial line (radius).

If OT was not perpendicular to PQ, then we can drop a perpendicular from O onto PQ and call the foot of the perpendicular S. Thus we see that OS (the perpendicular) is shorter than OT (some other line joining O and a point on PQ). But is that possible?