4.6 Circle properties

Learning objectives

4.6A Circle terminologyRecognise the centre, radius, chord, diameter, circumference, tangent, arc, sector and segment of a circle.4.6B Chord and tangent properties of circles Two tangentsunderstand chord and tangent properties of circles Two tangents from a point to a circle are equal in length Tangents are perpendicular to the radius at the point of contact The line from the centre of a circle which is perpendicular to a chord bisects the chord (and the converse)4.6HA Intersecting chord propertiesunderstand and use the internal and external intersecting chord properties 4.6HB The term ‘cyclic quadrilateral’recognise the term ‘cyclic quadrilateral’4.6HC Circle angle propertiesunderstand and use angle properties of the circle including: (i) angle subtended by an arc at the centre of a circle is twice the angle subtended at any point on the remaining part of the circumference (ii) angle subtended at the circumference by a diameter is a right angle (iii) angles in the same segment are equal (iv) the sum of the opposite angles of a cyclic quadrilateral is 180° (v) the alternate segment theorem Formal proof of these theorems is not required

A circle is the set of points at a fixed distance from its centre. That fixed distance is the radius; a diameter is a chord through the centre and has length twice the radius.

Term	Meaning
circumference	the circle's boundary
chord	straight segment joining two points on the circle
tangent	line touching the circle at one point
arc	part of the circumference
sector	region between two radii and an arc
segment	region between a chord and its arc

A diameter is always a chord, but a chord is a diameter only when it passes through the centre.

A sector has two straight radius edges; a segment has one straight chord edge. Do not name either region from appearance alone.

A tangent is perpendicular to the radius at the point of contact. Two tangents drawn from the same external point have equal lengths.

Condition	Consequence
centre-to-chord line is perpendicular	it bisects the chord
centre-to-chord line bisects the chord	it is perpendicular to the chord
equal chords	they are equally distant from the centre

Add the radius to a tangent diagram to create a right angle. Join the centre to a chord midpoint to create two congruent right triangles when useful.

The right angle is between the tangent and the radius at the contact point, not between a tangent and every chord through that point.

For chords $AB$ and $CD$ intersecting inside a circle at $X$ , $AX imes XB=CX imes XD$ .

From an external point $P$ , if two secants meet the circle at $A,B$ and $C,D$ , then $PA imes PB=PC imes PD$ , using each external length times its whole secant length.

Step	Action
1	identify the common intersection point
2	label the two parts of each chord or the external and whole secant lengths
3	equate the two products
4	solve and reject impossible negative lengths

For an external secant, the second factor is the whole length from the external point to the far circle intersection, not just the portion inside the circle.

A cyclic quadrilateral has all four vertices on one circle. The circle is its circumcircle.

Evidence	Conclusion
four vertices lie on one circle	cyclic
a pair of opposite angles sums to $180^\circ$	cyclic
an exterior angle equals the opposite interior angle	cyclic

The sides of a cyclic quadrilateral are chords of the circle; its diagonals are also chords.

A quadrilateral drawn inside a circle is not necessarily cyclic: every vertex must lie on the circumference, not merely inside the disk.

Angles subtended by the same chord at the circumference are equal. The angle at the centre is twice the angle at the circumference standing on the same arc, and an angle in a semicircle is $90^\circ$ .

Configuration	Result
cyclic quadrilateral	opposite angles sum to $180^\circ$
tangent and chord	angle between them equals the angle in the alternate segment
two radii	they form an isosceles triangle

If chord $AC$ subtends $38^\circ$ at point $B$ , then angle $AOC=76^\circ$ . Since $OA=OC$ , each base angle in triangle $AOC$ is $(180-76)/2=52^\circ$ .

Mark the chord or arc each angle stands on before selecting a theorem, then combine with triangle, straight-line or point-angle facts.

The centre angle is double only when both angles subtend the same arc. Do not double merely because one angle is drawn near the centre.

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—4.7 Geometrical reasoningRead

4.6 Circle properties

Learning objectives

Recognise circle terminology

Use chord and tangent properties

Use internal and external intersecting chord properties

Recognise cyclic quadrilaterals

Use circle angle theorems

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