Tangents to a circle

Lay a ruler across a coin and slide it. A straight line and a circle can meet in three ways:

The word tangent comes from the Latin tangere, 'to touch'. The Danish mathematician Thomas Fineke first used it in 1583. The single common point is the point of contact.

Tangents in daily life: a rope over a pulley above a well leaves the wheel along a tangent, and a bicycle wheel touches the road at just one point.

A tangent is a secant whose two points have come together. Push a secant away from the centre, keeping it parallel. Its two meeting points slide towards each other, the chord between them gets shorter, and at last the two points become one. Now the line is a tangent.

How many tangents? Exactly one at each point of the circle, so a circle has infinitely many. At most two tangents can be parallel to a given line: one on each side.

Theorem 10.1: the tangent is perpendicular to the radius at the point of contact. Why? Take the tangent at P. Every other point Q on it lies outside the circle, so OQ is longer than the radius OP. So OP is the shortest way from O to the line, and the shortest way to a line is always the perpendicular.

This gives you a right angle for free. Whenever a radius meets a tangent, mark 90°. Then use Pythagoras or trigonometry.