Circles

Tangents from an outside point

Use the fact that two tangents from an outside point are equal.

How many tangents can you draw from a point P? It depends on where P is:

P is	Tangents through P
inside the circle	none: every line through P cuts the circle twice
on the circle	exactly one
outside the circle	exactly two

The length of a tangent is the distance from the outside point to its point of contact.

Theorem 10.2: the two tangents from an outside point are equal. If PQ and PR touch the circle at Q and R, then PQ = PR.

Proof (Pythagoras): both angles at Q and R are 90° (Theorem 10.1). So PQ² = OP² – OQ² and PR² = OP² – OR². Since OQ = OR (radii), PQ = PR.

(You can also prove it with RHS congruence: OQ = OR, OP is shared, and both triangles have a right angle.)

The same congruence gives more: OP cuts in half the angle between the two tangents, and the tangents make equal angles at the centre.

A habit for every figure: find each outside point and mark its two equal tangents first. Most questions are then just adding and taking away.

Two circles with the same centre: a chord of the big circle that touches the small one is cut in half at the point of contact. The radius to that point is perpendicular to the chord, and a perpendicular from the centre always halves a chord. With radii 5 and 3: half the chord is √(25 – 9) = 4, so the chord is 8 cm.

From a point outside a circle you can draw exactly two tangents, and they are always equal in length.

The two tangents from an outside point are equal.

Practise this lesson

Questions that get harder as you improve, a Tutor that helps without giving the answer away, and progress you can see. Free for every learner.

Try it now Sign up free I have an account

No account, no email address. Opens the real product.