Introduction
Circle theorems in SSC exams are almost entirely applications of a handful of standard results — memorise
these, and most circle geometry questions become one- or two-step calculations.
Chord Properties
| Property |
|---|
| A perpendicular from the centre to a chord bisects the chord |
| Equal chords are equidistant from the centre |
| Equal chords subtend equal angles at the centre |
Tangent Properties
| Property |
|---|
| A tangent is perpendicular to the radius at the point of contact |
| Lengths of two tangents drawn from an external point to a circle are equal |
Q. Two tangents from an external point P touch a circle at A and B. If PA = 8 cm, find PB.
Tangents from the same external point are equal → PB = 8 cm
Angle Theorems
Flowchart — Key Angle Theorems
Angle subtended by an arc at the centre = 2 × angle subtended at any point on the remaining circumference
↓
Angle in a semicircle = 90° (special case of the above, since the centre-angle for a diameter is 180°)
↓
Angles in the same segment of a circle are equal
Q. An arc of a circle subtends an angle of 70° at the centre. Find the angle subtended by the same arc at a point on the remaining part of the circumference.
Angle at circumference = ½ × angle at centre = ½ × 70° = 35°
Cyclic Quadrilateral
A quadrilateral whose all four vertices lie on a circle. Opposite angles of a cyclic quadrilateral are supplementary (sum = 180°).
Q. In a cyclic quadrilateral ABCD, angle A = 75°. Find angle C.
Opposite angles supplementary → C = 180° − 75° = 105°
Alternate Segment Theorem
💡 Key Result: The angle between a tangent and a chord drawn from the point of contact equals the angle in the alternate segment (the angle subtended by that chord on the opposite arc).
Tangent-Secant Relationships
| Relationship |
|---|
| If PT is a tangent and PAB a secant from external point P: PT² = PA × PB |
Q. From an external point P, a tangent PT = 12 cm and a secant PAB is drawn where PA = 8 cm. Find PB.
PT² = PA × PB → 144 = 8 × PB → PB = 144/8 = 18 cm
✅ Practice Focus: Equal tangents from an external point · Centre-angle = 2 × circumference-angle rule · Angle in a semicircle = 90° · Cyclic quadrilateral's supplementary opposite angles · Tangent-secant length relationship (PT² = PA×PB).