Why do we care about a line that just kisses a circle?
That line – called a tangent – shows up in many exam problems. Knowing its tricks can save you minutes.
💡 In Simple Words: A tangent touches a circle at exactly one point. At that spot, the line and the circle are like a perfect handshake – the line doesn’t cut through, it just meets.
What exactly is a tangent?
A tangent (first time: a straight line that meets a circle at only one point) is a special line. If you picture water flowing through a pipe, the tangent is like a tiny leak that touches the pipe wall at just one spot and then runs away.
The first big theorem – radius ⟂ tangent
The line drawn from the centre of the circle to the point where the tangent meets is always perpendicular (means at a 90° angle) to the tangent.
Why? Imagine the radius as a stick stuck straight into the circle. If the tangent tried to poke into the circle, the stick would have to bend – impossible for a straight line. So the only way they can just touch is when the stick points straight out, making a right angle.
Proof (simple version):
- Let O be the centre, P the point of contact, and let the tangent be line l.
- Draw any other line from O to a point Q on l (Q ≠ P). Triangle OQP is formed.
- Because OP is a radius, OP = OQ (both are radii). So triangle OQP is isosceles with OP = OQ.
- In an isosceles triangle, the altitude from the vertex to the base also bisects the base and forms a right angle. Here, the altitude is exactly the line OP because P is the only point where l meets the circle.
- Hence ∠OPL = 90°.
The second big theorem – equal tangents from an external point
If you stand outside the circle at point A and draw two tangents AP and AQ, those two line segments have the same length.
Think of it like two identical ropes stretched from the same pole to touch a fence at two different spots – the ropes end up the same length because the fence is perfectly round.
Proof (quick sketch):
- Join the centre O to A, and also to the points of contact P and Q.
- Now you have two right‑angled triangles: ΔOPA and ΔOQA (right angle because of the first theorem).
- Both triangles share the side OA and have OP = OQ (radii).
- By the hypotenuse‑leg (HL) congruence rule – if the hypotenuse and one leg of two right triangles are equal, the triangles are identical.
- Therefore AP = AQ.
Worked example – finding a tangent length
Given: Circle centre O(0,0) radius 5 cm. External point A(13,0). Find the length of the tangent from A to the circle.
- Step 1: Compute OA = distance between (0,0) and (13,0) = 13 cm.
- Step 2: Use the right‑triangle relation: OA² = r² + (tangent)².
- Step 3: Plug in numbers: 13² = 5² + t² → 169 = 25 + t² → t² = 144.
- Step 4: t = √144 = 12 cm.
So the tangent from A touches the circle 12 cm away.
Quick comparison table
| Theorem | Statement | Key Idea in Proof |
|---|---|---|
| Radius ⟂ Tangent | The radius drawn to the point of contact is perpendicular to the tangent. | Isosceles triangle → altitude = perpendicular. |
| Equal Tangents | Tangents drawn from the same external point are equal in length. | Two right triangles share hypotenuse and a leg → HL congruence. |
Bullet‑point cheat sheet
- Touch‑only point → tangent.
- Radius at touch point ⟂ tangent.
- From one outside point, both tangents have same length.
- Use Pythagoras: (external distance)² = (radius)² + (tangent)².
- Remember: right‑triangle tricks (HL, Pythagoras) solve most problems.
📝 Likely Exam Questions
- State and prove the theorem that a radius drawn to the point of contact of a tangent is perpendicular to the tangent.
Answer: (Provide statement, draw diagram, use isosceles triangle argument as above.) - From a point P(8,‑3) two tangents are drawn to the circle x²+y²=25. Find the length of each tangent.
Answer: Compute OP distance, apply OA² = r² + t² → t = √(OP²‑25). - Explain why two tangents drawn from an external point to a circle are equal.
Answer: Show two right triangles OPA and OQA are congruent by HL, giving AP = AQ. - In a diagram, the centre O, external point A and points of contact B and C are given. If OA = 13 cm and radius = 5 cm, find AB.
Answer: Same as worked example, AB = 12 cm. - Prove that the angle between two tangents drawn from an external point equals the difference between 180° and the central angle subtended by the points of contact.
Answer: Use the fact that each tangent is perpendicular to its radius, form quadrilateral OBC, apply linear pair angles.