Circles — Tangent and Secant Properties
Introduction
A tangent is a line that touches the circle at exactly one point (the point of tangency). A secant is a line that intersects the circle at two distinct points. In ICSE Class 10, you study the properties of tangents and the relationships between secants and tangents drawn from an external point.
Key Definitions
- Tangent — A line that touches the circle at exactly one point.
- Point of contact — The point where the tangent touches the circle.
- Secant — A line that cuts the circle at two points.
- Common tangent — A tangent common to two circles (direct or transverse).
Tangent Properties
Property 1: Radius-Tangent Perpendicularity
The radius drawn to the point of contact of a tangent is perpendicular to the tangent.
If OT is the radius to the point of contact T, then OT ⟂ PT, where PT is the tangent.
Property 2: Lengths of Tangents from an External Point
The lengths of the two tangents drawn from an external point to a circle are equal.
If PA and PB are tangents from P to the circle, then PA = PB.
Property 3: Angle Between Tangents
The angle between two tangents drawn from an external point is supplementary to the angle subtended by the chord joining the points of contact at the centre.
∠APB + ∠AOB = 180°
Secant-Tangent Theorem
If a tangent and a secant are drawn from an external point P to a circle, then the square of the length of the tangent is equal to the product of the lengths of the secant and its external segment.
If PT is the tangent (with point of contact T) and PAB is the secant (A and B are intersection points), then:
PT² = PA × PB
Secant-Secant Theorem
If two secants PAB and PCD are drawn from an external point P to a circle, then:
PA × PB = PC × PD
Common Tangents to Two Circles
| Type | Condition | Description |
|---|---|---|
| Direct common tangent | Circles don't intersect | Both circles lie on same side of tangent |
| Transverse common tangent | Circles don't intersect | Tangent crosses the line joining centres |
Worked Examples
Example 1: Equal Tangents
From point P outside a circle, two tangents PQ and PR are drawn. If PQ = 7 cm, find PR.
Solution: Since lengths of tangents from an external point are equal: PR = PQ = 7 cm
Example 2: Radius-Tangent Perpendicularity
A tangent at point T makes an angle of 35° with chord AT. Find ∠ATO (O is the centre).
Solution: Since OT ⟂ PT (tangent), ∠OTP = 90° Given ∠ATP = 35° (angle between tangent and chord AT) ∠ATO = ∠OTP − ∠ATP = 90° − 35° = 55°
Example 3: Secant-Tangent Theorem
From an external point P, a tangent PT = 12 cm and a secant PAB where PA = 8 cm. Find AB.
Solution: By secant-tangent theorem: PT² = PA × PB 12² = 8 × PB 144 = 8 × PB PB = 18 cm
AB = PB − PA = 18 − 8 = 10 cm
Example 4: Secant-Secant Theorem
Two secants PAB and PCD are drawn from P. If PA = 6 cm, PB = 10 cm, PC = 5 cm, find PD.
Solution: By secant-secant theorem: PA × PB = PC × PD 6 × 10 = 5 × PD 60 = 5 × PD PD = 12 cm
Example 5: Tangents and Quadrilateral
A quadrilateral ABCD circumscribes a circle. Prove that AB + CD = BC + AD.
Solution: Let the circle touch AB at P, BC at Q, CD at R, and DA at S. Using equal tangents: AP = AS, BP = BQ, CQ = CR, DR = DS
AB + CD = (AP + PB) + (CR + RD) = (AS + BQ) + (CQ + DS) = (AS + DS) + (BQ + CQ) = AD + BC ✓
Comparison: Tangent vs Secant
| Feature | Tangent | Secant |
|---|---|---|
| Intersection points | Exactly one | Two |
| Length from external point | PT (tangent length) | PA, PB (segments) |
| Angle with radius | 90° at point of contact | Not fixed |
| Relation | PT² = PA × PB | PA × PB = PC × PD |
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| Applying secant-tangent theorem when PT is not a tangent | Verify PT is tangent (radius ⟂ tangent at point of contact) |
| Confusing secant lengths (PA vs PB) | PB = PA + AB (the full secant) |
| Forgetting equal tangent lengths property | Both tangents from same external point are equal |
| Assuming radius is perpendicular to any chord | Only the radius to the point of contact is perpendicular to the tangent |
ICSE Exam Focus
Tangent and secant properties carry 6–10 marks in ICSE exams. Questions include:
- Equal tangents and angle calculations.
- Secant-tangent theorem applications.
- Secant-secant theorem applications.
- Proof that a quadrilateral circumscribing a circle has equal sums of opposite sides.
- Constructions of tangents to a circle.
Marks Blueprint:
| Topic | Marks |
|---|---|
| Tangent properties (equal lengths, radius ⟂ tangent) | 3 |
| Secant-tangent theorem | 3 |
| Secant-secant theorem | 2 |
| Proofs | 3–4 |
Self-Test Questions
-
From point P outside a circle, PA and PB are tangents. If ∠APB = 60°, find ∠AOB and ∠OAB.
-
A tangent PT = 15 cm is drawn from P to a circle. A secant PAB passes through the circle with PA = 9 cm. Find AB.
-
Two secants PAB and PCD from P meet the circle at A, B and C, D respectively. If PA = 4 cm, AB = 5 cm, PC = 3 cm, find CD.
-
Prove that the lengths of tangents drawn from an external point to a circle are equal.
-
In a quadrilateral ABCD circumscribing a circle, prove that AB + CD = AD + BC.
In ICSE, the secant-tangent theorem (PT² = PA × PB) is a common 3-mark question. Always verify that the point of contact is correctly identified.
