Circle words you need first
A circle is all the points that are the same distance from one centre point O. That distance is the radius (r).
Put a straight line near a circle. Only three things can happen:
- 0 common points: the line is outside. It does not meet the circle.
- 2 common points: the line cuts the circle. It is called a secant.
- 1 common point: the line just touches. It is called a tangent. The touching point is the point of contact.
Easy test: let d be the distance from O to the line. If d > r, no point. If d < r, two points. If d = r, one point: a tangent.
Tangent at a point
Think of a secant cutting the circle at two points A and B. Now slowly slide it outward. A and B come closer and closer. At the last moment they join into one point. That line is the tangent.
How many tangents?
- At one point on the circle: exactly one tangent.
- From a point inside the circle: no tangent (every line through it cuts the circle twice).
- From a point outside the circle: exactly two tangents.
A circle can have endless tangents in total, one at each point on it.
Theorem 1: Tangent is perpendicular to the radius
Rule: the tangent at any point P of a circle is perpendicular (at 90°) to the radius OP.
Why? (proof in simple steps)
- Take any other point Q on the tangent, not P.
- Q is outside the circle (a tangent touches only at P). So OQ is longer than the radius: OQ > OP.
- This is true for every Q. So OP is the shortest distance from O to the line.
- The shortest line from a point to a line is always the perpendicular. So OP ⟂ tangent.
Use it: whenever you see a tangent and a radius meet, mark a right angle. Then you can use Pythagoras theorem.
Theorem 2: Two tangents from an outside point are equal
Rule: from a point T outside a circle, the two tangents TP and TQ have the same length.
Why? (proof in simple steps)
- Join OP, OQ and OT.
- ∠OPT = ∠OQT = 90° (Theorem 1).
- OP = OQ (both are radii) and OT is common.
- So triangles OPT and OQT are congruent (RHS rule).
- So TP = TQ. (CPCT: matching parts of congruent triangles are equal.)
Bonus facts from the same proof: OT cuts ∠PTQ into two equal halves, and ∠PTQ + ∠POQ = 180°.
Length formula: TP = √(OT² − r²).
Try it at home
Draw a circle with a bangle. Put a ruler so it just touches the circle at one point P. Draw the line. Find the centre by folding the paper circle twice. Join centre to P and measure the angle with a protractor: you will get 90°. Now pick a point T outside, draw both tangents and measure them. They will be equal.
Key formulas and definitions
- Tangent ⟂ radius at the point of contact: ∠OPT = 90°
- Tangents from an outside point: TP = TQ
- Tangent length: TP = √(OT² − r²)
- ∠PTQ + ∠POQ = 180°
- Line vs circle: d > r → 0 points, d = r → 1 point (tangent), d < r → 2 points (secant)
Worked examples
1. How many tangents can be drawn to a circle from a point on the circle?
Exactly one. At a point on the circle there is only one line that touches without cutting.
2. The radius is 5 cm and a point T is 13 cm from the centre. Find the length of the tangent from T.
The radius meets the tangent at 90°, so triangle OPT is right-angled at P. TP² = OT² − OP² = 13² − 5² = 169 − 25 = 144. So TP = 12 cm.
3. A tangent from T is 24 cm long and OT = 25 cm. Find the radius.
r² = OT² − TP² = 625 − 576 = 49. So r = 7 cm.
4. Two tangents TP and TQ are drawn from T. ∠PTQ = 70°. Find ∠POQ.
In quadrilateral OPTQ, ∠OPT = ∠OQT = 90°. Angles add to 360°. So ∠POQ = 360° − 90° − 90° − 70° = 110°.
5. TP and TQ are tangents from T, and ∠PTQ = 50°. Find ∠TPQ.
TP = TQ, so triangle TPQ is isosceles. ∠TPQ = ∠TQP = (180° − 50°) ÷ 2 = 65°.
6. A quadrilateral ABCD is drawn around a circle (all four sides touch it). AB = 6 cm, BC = 7 cm, CD = 4 cm. Find AD.
From each corner the two tangents are equal. Adding them gives AB + CD = BC + AD. So 6 + 4 = 7 + AD, and AD = 3 cm.
7. Two circles have the same centre. Radii are 5 cm and 3 cm. Find the length of a chord of the big circle that touches the small circle.
The chord is a tangent to the small circle, so the small radius meets it at 90° at its middle point. Half chord = √(5² − 3²) = √16 = 4 cm. Full chord = 8 cm.
Common mistakes
- Thinking a tangent can touch at two points. A tangent touches at exactly one point; a line meeting at two points is a secant.
- Forgetting to mark the 90° angle where radius and tangent meet. That right angle is the key to almost every sum.
- Putting OT as a leg in Pythagoras. OT (centre to outside point) is the hypotenuse: TP² = OT² − r².
- Saying a point inside the circle has one tangent. A point inside has no tangent at all.