CARA'S TIPSCircle geometry theorems
JAMBWAECNECO
What the syllabus asks
JAMB: Angles in a circle, cyclic quadrilaterals, tangents and chords
WAEC: Circle geometry: parts of a circle; chord properties; angle subtended at the centre is twice that at the circumference; angles in the same segment; angle in a semicircle; cyclic quadrilaterals; tangent properties (tangent ⟂ radius, equal tangents from an external point); alternate segment theorem; intersecting chords and tangent–secant results; proofs and riders
NECO: Circle geometry: parts of a circle; chord properties; angle subtended at the centre is twice that at the circumference; angles in the same segment; angle in a semicircle; cyclic quadrilaterals; tangent properties (tangent ⟂ radius, equal tangents from an external point); alternate segment theorem; intersecting chords and tangent–secant results; proofs and riders
What you will learn
- Name the parts of a circle.
- Use chord properties to calculate lengths.
- Apply the angle theorems: centre/circumference, same segment, semicircle.
- Apply cyclic quadrilateral properties.
- Apply tangent properties and the alternate segment theorem.
- Use intersecting chord and tangent–secant results; prove simple theorems and solve riders with reasons.
Key points
- Perpendicular from centre bisects a chord; use half-chord with Pythagoras.
- Angle at centre = 2 × angle at circumference on the same arc.
- Angles in the same segment are equal; angle in a semicircle = 90°.
- Cyclic quadrilateral: opposite angles sum to 180°; exterior = interior opposite.
- Tangent ⟂ radius; tangents from an external point are equal.
- Alternate segment: tangent–chord angle = angle in alternate segment.
- AX × XB = CX × XD; PT² = PA × PB.
- Always state reasons.
Start here: Parts of a circle
Think of a round bole (roasted plantain) seller's tray. The rim is the circumference; the centre O is the middle; a straight cut across is a chord; a cut through the middle is a diameter.
- Radius OA: centre to circumference. All radii of a circle are equal — this fact creates isosceles triangles everywhere in circle geometry.
- Arc: part of the circumference. The smaller is the minor arc, the larger the major arc.
- Segment: area cut off by a chord. Minor and major segments.
- Sector: "pizza slice" between two radii.
- Tangent: touches the circle at one point. Secant: cuts it twice.
In circle centre O, OA and OB are radii and ∠AOB = 100°. Find ∠OAB.
- OA = OB (radii), so triangle OAB is isosceles.
- Base angles equal: (180° − 100°)/2 = 40°.
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Also learn the difference between a sector and a segment: a sector is bounded by two radii, a segment by one chord.
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Practice questions
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Key words
- Circumference
- The boundary of a circle.
- Radius
- Line from centre to circumference.
- Diameter
- Chord through the centre; twice the radius.
- Chord
- Line joining two points on the circle.
- Arc
- Part of the circumference.
- Segment
- Region between a chord and an arc.
- Sector
- Region between two radii and an arc.
- Subtend
- To form an angle at a point by joining the ends of an arc or chord to that point.
- Cyclic quadrilateral
- Quadrilateral with all four vertices on a circle.
- Concyclic
- Lying on the same circle.
- Tangent
- A line touching the circle at one point.
- Secant
- A line cutting the circle at two points.
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