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)� = External segment � Whole secant

PT� = 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	PT� = 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