Why is the tangent to a circle always perpendicular to the radius at the point of contact?

The reason is that the radius drawn to the point of contact is the shortest distance from the centre to the tangent line. Among all line segments from the centre to a straight line, the perpendicular one is always the shortest. Since a tangent touches the circle at exactly one point and every other point on the tangent lies outside the circle (farther from the centre than the radius), the point of contact must be the closest point. The closest point is reached by the perpendicular, so the radius at the point of contact is perpendicular to the tangent. This single fact unlocks many problems: whenever you see a tangent, immediately draw the radius to the contact point and mark a 90° angle there.