Two Tangents from a Point are Equal
From a point P outside a circle there are exactly two lines that touch it. They meet the circle at A and B, and the two tangent lengths are always equal: PA = PB.
The shape OAPB is a kite. OA and OB are radii so they are equal, PA and PB are the tangents so they are equal, and OP is the line of symmetry running through both.
How to Use
Interactive Exploration
Drag P towards or away from the circle. Both tangents lengthen and shorten together, and the points of contact slide round to follow. Bring P in close and the tangents shrink; push it out and they stretch, but they never differ from each other.
Drag A or B instead to swing the whole kite around the circle without changing its size.
The right angle at each point of contact is the tangent and radius theorem doing the work — it is what makes the two triangles congruent.
Solving Problems
New positions sets a fresh question: one tangent keeps its length and the other is marked x. Tap the x on the diagram, or click PA or PB, to check.
The Style tab changes the colours and the background paper.
Teaching Notes
- P is placed by the tangent length rather than by its distance from the centre, so the answers land on half units instead of awkward decimals.
- Why it works: triangles OAP and OBP share the hypotenuse OP, have equal radii OA and OB, and are both right-angled at the point of contact. That is enough to make them congruent (RHS), so PA = PB.
- This also means OP bisects both the angle at P and the angle at the centre — worth asking about once the length result is secure.
- A useful follow-up: what happens as P moves onto the circle itself? The two tangents merge into one and the length drops to zero.