Circle theorems II.
Use symmetry to connect chord lengths, centre distances and equal tangent segments.
Use equal-chord distances · apply the chord bisector property · calculate with equal tangents
Equal chords sit equally far from the centre
Chords connect two points on a circle, and their positions are related to the centre. In the same circle, equal chords sit at equal perpendicular distances from the centre.
The word perpendicular matters: an arbitrary slanted segment from the centre is not the distance to a line. Draw or identify a right angle before comparing those distances. Keep “same circle” or equal-radius circles in view when applying equal-chord relationships.
In the same circle, equal chords are equidistant from the centre. Measure that distance along the perpendicular from the centre to each chord.
Equal chords AB and CD are 4.5 cm and d cm from the centre. Find d.
d = 4.5 cm. Reason: equal chords are equidistant from the centre.
Pause and explainWhich distance from the centre to a chord is used in the theorem?
The perpendicular distance, which is the shortest distance from the centre to the chord’s line.
A chord’s perpendicular bisector passes through the centre
Joining the centre to the endpoints of a chord creates an isosceles triangle, since both sides are radii. A perpendicular from the centre divides that triangle into two matching right-angled triangles.
This explains why the perpendicular bisects the chord. It also explains why a line through the centre and midpoint of a chord is perpendicular to the chord. Once the half-chord is known, Pythagoras can connect it with the radius and centre-to-chord distance.
The perpendicular from the centre to a chord bisects it. Equivalently, the chord’s perpendicular bisector passes through the centre of the circle.
A circle has radius 10 cm and a chord of length 16 cm. Find the distance from its centre to the chord.
The perpendicular from the centre meets the chord halfway, so half the chord is 8 cm. The radius is the hypotenuse of a right triangle.
d = √(100 − 64) = 6 cm
Pause and explainA perpendicular from the centre meets a 12 cm chord. What is each half of the chord?
6 cm, because that perpendicular bisects the chord.
Tangents from one external point are equal
A tangent touches a circle at one point. Drawing two tangents from the same external point creates equal tangent segments to the two points of contact.
Join the centre to each contact point to reveal right angles with the tangents. These extra radii help you see matching right-angled triangles and connect the tangent-length fact with Pythagoras or angle sums. The equal segments must start from the same external point.
PA and PB are tangents from the same point P. PA = 3x + 2 and PB = 5x − 8. Find x and the tangent lengths.
x = 5
PA = PB = 17
Reason: tangents from the same external point are equal in length.
OP = 10 cm and the radius OA = 6 cm. PA is tangent at A. Find PA.
Radius OA is perpendicular to tangent PA, so triangle OAP is right-angled.
Pause and explainAre any two tangent segments to a circle automatically equal?
No. The equal-tangent result applies to segments from one common external point.
Your turn. Justify the symmetry.
Add radii and perpendiculars to your sketch where they expose the needed geometry. Mark half-chords rather than confusing them with full chord lengths.
Before equating tangent lengths, identify their common external point. When calculating a distance, state the right-angled triangle being used and which side is its radius.
A chord is 24 cm long in a circle of radius 13 cm. Find its perpendicular distance from the centre.
Reveal worked answer
Two equal chords lie 7 cm and x cm from the centre, measured perpendicularly. Find x.
Reveal worked answer
Two tangent segments from P have lengths 4x − 1 and 2x + 9. Find x.
Reveal worked answer
OP = 13 cm and a tangent from P touches a circle of radius 5 cm. Find the tangent length.