Chords and tangents | Year 11

01 / LEARN

The idea, step by step

Chords connect length to distance

The perpendicular from a circle centre to a chord bisects the chord. Joining the centre to the chord endpoints creates congruent right triangles: their hypotenuses are radii and they share a perpendicular leg. Equal chords are equally distant from the centre; the converses also hold within one circle or equal-radius circles.

If a chord has half-length a and distance d from the centre of a circle with radius r, then $a^{2} + d^{2} = r^{2}$ . Use half the chord length, not its full length, in this right triangle.

Tangents have two useful angle facts

A tangent is perpendicular to the radius at the point of contact. Two tangent segments drawn from the same external point have equal lengths, which follows from congruent right triangles with a shared centre-to-point hypotenuse and equal radii.

The angle between a tangent and a chord equals the inscribed angle in the alternate segment. Identify the chord and the point on the opposite segment before matching the angles. Equal tangent segments do not mean every secant segment from that point has the same length.

A common trap

Using the whole chord as the short leg of the radius triangle.

Instead: The centre perpendicular bisects the chord; use its half-length in Pythagoras.

02 / WORKED EXAMPLES

See the thinking

Example 1

A chord from its centre distance

A circle has radius 13 cm. A chord is 5 cm from the centre. Find its length.

The perpendicular from the centre bisects the chord. Let its half-length be a.
$a^{2} + 5^{2} = 13^{2}$ , so $a^{2} = 144$ and $a = 12$ .
The full chord length is $2 a$ .

24 cm.

Example 2

Two tangents

PA and PB are tangent segments from P to a circle with centre O. If PA is 15 cm and radius OA is 8 cm, find PB and OP.

Equal tangents from P give $P B = P A = 15$ cm.
OA is perpendicular to PA, so triangle OAP is right-angled.
$O P^{2} = 8^{2} + 15^{2} = 289$ .

= 15

cm and OP

= 17

cm.

Check your understanding

Two tangents from P have lengths $3 x + 1$ and $5 x - 7$ . Find x.

Show answer

$x = 4$ .

Set the lengths equal: $3 x + 1 = 5 x - 7$ , so $2 x = 8$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

A radius meets a tangent at its point of contact. Find the angle between them.
2 marks
A radius-5 cm circle has a chord 3 cm from its centre. Find the chord length.
2 marks
Two tangent lengths from P are $2 x + 7$ and $4 x - 5$ . Find x and their common length.
3 marks
A tangent and chord AB make a $52^{\circ}$ angle. C is on the alternate segment. Find angle ACB.
2 marks
Two chords in the same circle are each 16 cm long. One is 6 cm from the centre. How far is the other from the centre? Explain.
2 marks
A tangent PT is 24 cm long and OP is 25 cm, where O is the circle centre. Find the radius OT.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A radius meets a tangent at its point of contact. Find the angle between them.
2 marks
The tangent-radius theorem gives perpendicular lines.
$90^{\circ}$ .
A radius-5 cm circle has a chord 3 cm from its centre. Find the chord length.
2 marks
Half-chord $a = \sqrt{25 - 9} = 4$ ; double it.
8 cm.
Two tangent lengths from P are $2 x + 7$ and $4 x - 5$ . Find x and their common length.
3 marks
Set $2 x + 7 = 4 x - 5$ , so $x = 6$ ; substitute.
$x = 6$ ; 19 units.
A tangent and chord AB make a $52^{\circ}$ angle. C is on the alternate segment. Find angle ACB.
2 marks
Apply the alternate-segment theorem to chord AB.
$52^{\circ}$ .
Two chords in the same circle are each 16 cm long. One is 6 cm from the centre. How far is the other from the centre? Explain.
2 marks
Equal chords in one circle are equally distant from the centre.
6 cm.
A tangent PT is 24 cm long and OP is 25 cm, where O is the circle centre. Find the radius OT.
3 marks
OT is perpendicular to PT.
$O T^{2} = 25^{2} - 24^{2} = 49$ , so $O T = 7$ cm.
7 cm.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: geometric reasoning and real-number/proof material, regrouped into a school chapter; all exercises are original.

Official curriculum source ↗

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