E4.8 Circle theorems II

Learning objectives

E4.8.1Use the following symmetry properties of circles: • equal chords are• Use the following symmetry properties of circles: • equal chords are equidistant from the centre • the perpendicular bisector of a chord passes through the centre • tangents from an external point are equal in length. Candidates are expected to use the geometrical properties listed in the syllabus when giving reasons for answers.

A circle is symmetric about every line through its centre, so matching chords or tangents create equal lengths and perpendicular bisectors that can be used as exact geometric reasons.

Configuration to recognise	Property to use	Useful conclusion
two equal chords in the same circle	equal chords are equidistant from the centre	perpendicular distances from the centre to the chords are equal
a chord and its perpendicular bisector	the perpendicular bisector of a chord passes through the centre	joining the centre to the chord's midpoint gives a right angle to the chord
two tangents drawn from one external point	tangents from an external point are equal in length	the two tangent segments form an isosceles triangle

For a chord, the perpendicular from the centre splits it into two equal halves: the two right triangles contain equal radii and matching half-chords, so the distance from the centre is fixed. For tangents $TP$ and $TQ$ , radii $OP$ and $OQ$ are perpendicular to the tangents; right triangles $OPT$ and $OQT$ share hypotenuse $OT$ and have $OP=OQ$ , so congruence gives $TP=TQ$ .

Mark the centre, chord midpoints, perpendicular signs, radii, tangent contact points and the common external point. State the circle property first, then use congruence, Pythagoras or ordinary angle facts only as separate justified steps.

In one circle, equal chords $AB$ and $CD$ have perpendicular distances $OM=5$ cm and $ON$ from centre $O$ . Then $ON=5$ cm because equal chords are equidistant from the centre. If tangents from $T$ touch at $P$ and $Q$ and $TP=8$ cm, then $TQ=8$ cm because tangents from the same external point are equal.

Distance from the centre to a chord means the perpendicular distance, not a sloping segment to an endpoint. Tangent lengths are equal only when both tangents start from the same external point, and a chord's perpendicular bisector—not every perpendicular line—must pass through the centre.

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—Cambridge IGCSE Mathematics (0580)Read