Circle Theorems

Circle Theorems

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

A circle theorem links angles, arcs, chords, tangents and segments formed by lines meeting at, inside or outside a circle. Six rules cover roughly 90% of NABTEB questions:

1. Angle at centre = 2 × angle at circumference on the same arc.
2. Angle in a semicircle = 90° (Thales’ theorem).
3. Angles in the same segment are equal.
4. Opposite angles of a cyclic quadrilateral sum to 180°.
5. Exterior angle of a cyclic quadrilateral = interior opposite angle.
6. Tangent–chord angle = angle in the alternate segment.

Always identify the arc first, then pick which rule applies. For NABTEB Paper 1, one of these six facts is the key to nearly every geometry MCQ. Watch for the reflex angle trap: when an angle at the centre looks bigger than 180°, use 360° − reflex to get the non-reflex arc.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Core Definitions

• Chord – a straight line joining two points on the circle.
• Arc – part of the circumference between two points (major or minor).
• Tangent – a line that touches the circle at exactly one point.
• Secant – a line cutting the circle at two points.
• Cyclic quadrilateral – a four-sided figure whose four vertices all lie on the circle.

The Seven Essential Theorems

Theorem 1 – Angle at the Centre: The angle subtended by an arc at the centre is twice the angle subtended at any point on the remaining arc of the circle. If arc PQ subtends ∠POQ at centre O and ∠PRQ on the circumference, then ∠POQ = 2∠PRQ.

Theorem 2 – Angle in a Semicircle: Any angle inscribed in a semicircle (with the diameter as the base) is a right angle. This is a special case of Theorem 1.

Theorem 3 – Angles in the Same Segment: All angles subtended by the same chord, standing on the same arc, are equal.

Theorem 4 – Cyclic Quadrilateral: Opposite angles of a cyclic quadrilateral sum to 180° (they are supplementary).

Theorem 5 – Exterior Angle of Cyclic Quadrilateral: An exterior angle equals the interior angle at the opposite vertex.

Theorem 6 – Tangent–Chord (Alternate Segment Theorem): The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment (the segment on the opposite side of the chord).

Theorem 7 – Tangent–Radius: A tangent is perpendicular to the radius at the point of contact. Two tangents from an external point are equal in length.

Worked Relationship

In circle with centre O, chord PQ subtends ∠POQ = 140° at the centre. Angle PRQ on the major arc = 140°/2 = 70°, and any angle on the minor arc = 180° − 70° = 110°.

NABTEB Question Patterns

• Calculate an unknown angle given a diagram with centre, tangent, chord, or cyclic quadrilateral.
• State which theorem justifies a given angle equality.
• Find angles x and y simultaneously using two theorems in one figure.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Edge Cases and the Reflex-Angle Trap

When the central angle given is the reflex angle (greater than 180°), students often apply Theorem 1 directly and double a wrong value. The correct arc measure for the inscribed-angle rule is always the non-reflex central angle. If ∠POQ = 240° (reflex), the inscribed angle on the major arc uses 360° − 240° = 120°, giving inscribed angle = 60°.

Another trap: Theorem 3 only applies when both angles stand on the same segment. If one vertex sits on the minor arc and the other on the major arc, the two angles are supplementary, not equal.

Connection to Coordinate Geometry

Circle theorems link directly to the equation of a circle, x² + y² = r². The tangent–radius perpendicularity (Theorem 7) gives the gradient of a tangent as the negative reciprocal of the radius gradient — useful when NABTEB Paper 2 mixes pure and coordinate geometry.

Common Mistakes

• Treating a tangent–chord angle as equal to the angle in the same segment instead of the alternate segment.
• Applying Theorem 1 to a chord that does not pass through the given centre point.
• Forgetting that a cyclic quadrilateral requires all four vertices on the circle; otherwise Theorem 4 fails.
• Using 360° instead of 180° when checking opposite angles of a cyclic quadrilateral.

Practice Prompts

1. In a circle, chord AB subtends ∠AOB = 108° at centre O. Find (i) the angle ACB where C is on the major arc, (ii) the angle ADB where D is on the minor arc.
2. PQRS is a cyclic quadrilateral with ∠P = 64°. Find ∠Q, ∠R and the exterior angle at S.

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