Ever wondered why a line that just kisses a circle feels so special? That's a tangent, and it hides some neat geometry tricks.
A tangent is a straight line that touches a circle at exactly one point. The magic? At that point, the line is perfectly perpendicular to the radius, which is the line from the centre of the circle to the touch point.
What is a Tangent to a Circle?
Think of a circle as a round pond and the radius as a rope tied to the centre. If you pull the rope straight out until it just barely touches the edge, the rope is now a tangent. The point where they meet is called the point of contact. The key rule: the radius meets the tangent at a right angle (90°).
Key Theorems About Tangents
1. The Tangent‑Radius Perpendicular Theorem
**Statement**: The radius drawn to the point of contact of a tangent is perpendicular to the tangent.
Why it works: Imagine the centre, the point of contact, and any other point on the tangent forming a triangle. The shortest distance from the centre to the line is the radius, so it must hit the line at a right angle.
2. Tangents from an External Point are Equal
**Statement**: If you pick a point outside the circle and draw two tangents to the circle, the lengths of those two tangents are the same.
Proof sketch:
- Let the external point be P and the touch points be A and B.
- Join P to the centre O, and draw radii OA and OB.
- Triangles OPA and OPB share OP and have OA = OB (both are radii).
- Both triangles are right‑angled at A and B (by the first theorem).
- By the hypotenuse‑leg (HL) congruence rule, the triangles are congruent, so PA = PB.
3. Angle Between Two Tangents
**Statement**: The angle formed between two tangents drawn from an external point equals the difference between 180° and the central angle subtended by the points of contact.
In simpler words, if you know the angle at the centre between the two touch points, you can find the angle outside the circle.
Proof outline:
- Draw radii to the two points of contact, creating triangle OPA and OPB.
- Each radius is perpendicular to its tangent, giving two right angles.
- The quadrilateral formed (A‑O‑B‑P) is cyclic, allowing us to use the property that opposite angles sum to 180°.
- Rearrange to get the desired relationship.
4. Tangent‑Chord Angle Theorem
**Statement**: The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment (the angle subtended by the chord at any point on the opposite side of the circle).
This one is handy for solving many exam problems where a chord and a tangent appear together.
Proof sketch:
- Let the chord be AB and the tangent at A be AT.
- Draw the diameter through A to point D.
- Triangle ABD is inscribed, so angle ABD equals angle ACD (same chord AD).
- Since AT is perpendicular to radius at A, angle between AT and AB equals angle in the opposite segment.
Worked Example
**Problem**: From a point P outside a circle with centre O, two tangents PA and PB are drawn. If OP = 13 cm and the radius OA = 5 cm, find the length of each tangent.
Solution**:
- Draw OA and OB. Both are 5 cm.
- Triangle OPA is right‑angled at A (tangent‑radius theorem).
- Use Pythagoras: OP² = OA² + PA² → 13² = 5² + PA².
- Compute: 169 = 25 + PA² → PA² = 144 → PA = 12 cm.
- By the equal‑tangents theorem, PB = PA = 12 cm.
So each tangent measures 12 cm.
Quick Comparison Table
| Theorem | Key Idea | Typical Use in Exams |
|---|---|---|
| Tangent‑Radius Perpendicular | Radius ⟂ Tangent at point of contact | Identify right angles, set up Pythagoras |
| Equal Tangents from External Point | PA = PB | Find unknown lengths, prove congruence |
| Angle Between Two Tangents | ∠APB = 180° – ∠AOB | Calculate unknown angles |
| Tangent‑Chord Angle | ∠(tangent, chord) = angle in opposite segment | Relate tangent‑chord angle to interior angles |
Common Mistakes to Avoid
- Assuming the tangent always passes through the centre – it never does.
- Mixing up the external angle formula with the interior central angle.
- Forgetting that the equal‑tangents rule only works when both lines truly touch the circle.
📝 Likely Exam Questions
- Question: Prove that the two tangents drawn from an external point to a circle are equal in length.
Answer: Connect the external point to the centre, draw radii to the points of contact, note the right angles, and apply the HL congruence rule to the two right‑angled triangles. - Question: In a circle, the radius is 7 cm and a tangent from an external point is 24 cm long. Find the distance from the external point to the centre.
Answer: Use Pythagoras: OP² = OA² + PA² → OP = √(7² + 24²) = √(49+576) = √625 = 25 cm. - Question: The angle between two tangents drawn from point P to a circle is 50°. What is the central angle subtended by the points of contact?
Answer: Central angle = 180° – 50° = 130°. - Question: A chord AB makes a 30° angle with the tangent at A. What is the angle subtended by AB at any point on the opposite arc?
Answer: By the tangent‑chord theorem, the angle in the alternate segment is also 30°. - Question: Show that if a line is perpendicular to a radius at its end point, the line is a tangent to the circle.
Answer: The line meets the circle at only one point (the radius end) and forms a right angle, satisfying the definition of a tangent.