Circles

ICSE Class 10 Mathematics • Chapter 11

1. Angle Properties of Circles

A. Central Angle vs Inscribed Angle

Central Angle: Angle at the center of the circle.

Inscribed Angle: Angle at any point on the circle (vertex on circumference).

Central Angle = 2 × Inscribed Angle

(if both subtend the same arc)

[Diagram: Circle with central angle ∠AOB and inscribed angle ∠ACB on same arc AB]

B. Angles in the Same Segment

Theorem: Angles in the same segment of a circle are equal.

All inscribed angles subtending the same arc are equal.

C. Angle in a Semicircle

Theorem: An angle inscribed in a semicircle is a right angle (90°).

If AB is a diameter and C is any point on the circle, then ∠ACB = 90°

2. Cyclic Quadrilateral

Cyclic Quadrilateral: A quadrilateral whose all four vertices lie on a circle.

Property 1: Opposite angles of a cyclic quadrilateral are supplementary.

∠A + ∠C = 180° and ∠B + ∠D = 180°

Property 2: Exterior angle of cyclic quadrilateral = Interior opposite angle.

[Diagram: Cyclic quadrilateral ABCD inscribed in a circle]

3. Tangent Properties

Tangent: A line that touches the circle at exactly one point (point of tangency).

Theorem 1: The tangent at any point of a circle is perpendicular to the radius at that point.

If PT is tangent at T, then OT ⊥ PT (∠OTP = 90°)

Theorem 2: Tangents drawn from an external point to a circle are equal in length.

If PA and PB are tangents from P, then PA = PB

[Diagram: Two tangents PA and PB from external point P, with PA = PB]

Additional Properties (Two Tangents from External Point):

PA = PB (equal tangent lengths)
∠OPA = ∠OPB (OP bisects angle between tangents)
∠POA = ∠POB (OP bisects central angle)
OP passes through center and bisects AB at right angles

4. Intersecting Chords Theorem

Theorem: If two chords intersect inside a circle, then:

PA × PB = PC × PD

(Product of segments of one chord = Product of segments of other chord)

[Diagram: Two chords AB and CD intersecting at P inside circle]

5. Tangent-Secant Theorem

Theorem: If a tangent and a secant are drawn from an external point:

(Tangent)² = External segment × Whole secant

PT² = PA × PB

[Diagram: Tangent PT and secant PAB from point P]

6. Angle at Tangent (Alternate Segment Theorem)

Theorem: The angle between a tangent and a chord at the point of contact equals the inscribed angle in the alternate segment.

∠PTB = ∠BAT (angle in alternate segment)

7. Solved Examples

Example 1: In a circle, inscribed angle is 35°. Find the central angle subtending the same arc.

Central angle = 2 × Inscribed angle = 2 × 35° = 70°

Example 2: In cyclic quadrilateral ABCD, ∠A = 70° and ∠B = 115°. Find ∠C and ∠D.

∠C = 180° − ∠A = 180° − 70° = 110°

∠D = 180° − ∠B = 180° − 115° = 65°

Example 3: Two chords AB and CD intersect at P. If AP = 4, PB = 6, CP = 3, find PD.

AP × PB = CP × PD

4 × 6 = 3 × PD

PD = 8 cm

8. Quick Reference Table

Theorem	Statement
Central = 2 × Inscribed	∠at center = 2 × ∠at circumference
Same Segment	Angles in same segment are equal
Semicircle	Angle in semicircle = 90°
Cyclic Quadrilateral	Opposite angles add to 180°
Tangent ⊥ Radius	Tangent perpendicular to radius
Equal Tangents	Tangents from external point are equal
Intersecting Chords	PA × PB = PC × PD
Tangent-Secant	PT² = PA × PB

Exam Practice (PYQ Trends)

PYQ: 2023

BOARD In the figure, O is center of circle. ∠AOB = 100°. Find ∠ACB and ∠ADB if C and D are points on the major and minor arcs respectively.

PYQ: 2022

BOARD PA and PB are tangents to a circle with center O. If ∠APB = 80°, find: (i) ∠AOB (ii) ∠OAB