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Welcome to GCSE Edexcel Maths revision.

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Topic M 33: Circle theorems.

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This video covers Higher tier.

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Use circle theorems only when their conditions hold: identify the centre, the chord or arc involved, and points actually on the circumference.

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Give the theorem as the reason.

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The angle at the centre is twice the angle at the circumference standing on the same arc.

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An angle of 50 degrees at the circumference corresponds to 100 degrees at the centre.

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Choose the correct arc and centre angle.

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Centre angle twice circumference angle

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Angles in the same segment are equal: two circumference angles subtended by the same chord from the same side of it are equal.

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An angle in a semicircle is 90 degrees.

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A triangle with one side as a diameter and its other vertex on the circle is right-angled.

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Worked example: If A B is a diameter and C lies on the circumference, angle A C B equals 90 degrees.

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If angle C A B equals 32 degrees, then angle A B C equals 58 degrees by the triangle angle sum.

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A right angle subtended by a diameter

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Opposite angles of a cyclic quadrilateral sum to 180 degrees.

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All four vertices must lie on the circle; an arbitrary quadrilateral does not qualify.

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Opposite angles of a cyclic quadrilateral

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A tangent is perpendicular to the radius at the contact point.

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Mark the 90 degrees angle before using triangle-angle rules.

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A radius meeting a tangent at a right angle

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Tangents drawn from the same external point have equal lengths.

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If P A and P B touch the circle at A and B, P A equals P B.

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The perpendicular from the centre to a chord bisects the chord.

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If a radius and half-chord form a right triangle, Pythagoras can find their distances.

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The alternate segment theorem says the angle between a tangent and a chord equals the angle subtended by that chord in the opposite segment.

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Match the chord endpoints carefully.

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Alternate segment theorem

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Worked example: A cyclic quadrilateral has angle A equals 112 degrees.

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Its opposite angle C equals 180 degrees minus 112 degrees equals 68 degrees, by the cyclic quadrilateral theorem.

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Worked example: Two tangents from P meet at 60 degrees.

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Joining the centre O to both contacts creates two right angles.

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The angle between the radii is 360 degrees minus 90 degrees minus 90 degrees minus 60 degrees equals 120 degrees.

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To prove the centre-angle theorem in a suitable configuration,

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join radii to create isosceles triangles,

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use equal base angles and angle sums,

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then express the centre angle in terms of the circumference angle.

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For a proof with centre O inside triangle A C B, let angle A C O equals p and angle O C B equals q.

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Equal radii give angle A O C equals 180 degrees minus 2 p and angle B O C equals 180 degrees minus 2 q.

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Angles around O give the remaining angle A O B equals 2 multiplied by open bracket p plus q close bracket equals 2 angle A C B.

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Other placements require the corresponding angle subtraction.

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For equal tangents, triangles O A P and O B P are right-angled, have common hypotenuse O P and equal radii O A equals O B.

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R H S congruence gives P A equals P B.

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Do not rely on how an angle looks in a drawing.

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Use algebraic angle labels and a sequence of valid theorems for a proof; different placements may require adding rather than subtracting angles.

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That completes Circle theorems.

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Revisit the notes and test yourself on the revision website.