A Tangent Meets the Radius at 90°
A tangent is a line that touches a circle at exactly one point. Here it touches at T, and OT is the radius drawn to that point.
The angle between them is always 90°. Move T anywhere round the circle, or slide P along the tangent, and it does not change.
Because the three angles of triangle OTP add to 180°, the other two must add to 90° — which is what turns the theorem into something you can calculate with.
How to Use
Interactive Exploration
Drag T to move the point of contact. The tangent swings round with it, staying perpendicular to the radius. Drag P to slide it along the tangent, on either side of T.
Watch ∠TOP and ∠OPT trade off as you move P: as one grows the other shrinks, always summing to 90°. Here P is held between 25° and 50° at the centre, which keeps it on the grid — its distance along the tangent is r × tan(∠TOP), so it runs away quickly beyond that. Carried further in the imagination, the angle at the centre would approach 90° and the angle at P close towards nothing, but OP and the tangent would never actually become parallel.
Solving Problems
New positions sets a fresh question: one of the two acute angles is replaced by x while the other keeps its value. Subtract from 90°, then tap the x on the diagram or click ∠TOP or ∠OPT to check.
The Style tab changes the colours and the background paper.
Teaching Notes
- P is positioned by the angle it makes at the centre rather than by its distance along the tangent, so both acute angles are always whole numbers.
- Why it works: every other point on the tangent lies outside the circle, so OT is the shortest distance from O to the line — and the shortest distance from a point to a line is always the perpendicular.
- Worth connecting to the perpendicular to a chord: both are about a right angle at the centre, one to a line that cuts the circle and one to a line that only touches it.
- A common slip is assuming the right angle is at P or at O. Ask students where the 90° actually sits and why it has to be at the point of contact.