Chords and tangents | Year 11

01 / LEARN

Understand the idea.

Chords connect length to distance

The perpendicular from a circle centre to a chord bisects that chord. Join the centre to each endpoint: the two right triangles have equal radii as hypotenuses and share the perpendicular leg, so they are congruent.

Use the half-chord in Pythagoras: if its length is $a$ , its distance from the centre is $d$ and the radius is $r$ , then $a^{2} + d^{2} = r^{2}$ . Within one circle or equal-radius circles, equal chords have equal distances from the centre, and conversely.

Tangents have two useful angle facts

A radius to a point of tangency is perpendicular to the tangent. Two tangent segments from the same external point are equal: the centre, external point and contact points form congruent right triangles with equal radii and a shared hypotenuse.

The tangent–chord theorem relates the angle between a tangent and a chord to an inscribed angle in the alternate segment. Identify the chord and its opposite segment before matching angles. Tangent equality applies only to tangents from the same external point.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A chord from its centre distance

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

Worked solution

1
Define the half-chord.
$a = \frac{1}{2} \times chord length$
The perpendicular distance from the centre meets the chord at its midpoint.
2
Find the half-length using Pythagoras.
$\begin{aligned} a^{2} + 5^{2} & = 13^{2} \\ a^{2} & = 169 - 25 = 144 \\ a & = 12 cm \end{aligned}$
The radius is the hypotenuse; the half-chord and centre distance are perpendicular legs.
3
Recover the full length.
$chord length = 2 a = 24 cm$
The chord consists of two equal halves.

Answer

24 cm.

EXAMPLE 02

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.

Worked solution

1
Use equal tangents from P.
$P B = P A = 15 cm$
Tangent segments drawn from the same external point have equal lengths.
2
Identify the right triangle.
$O A ⟂ P A$
The radius to the contact point is perpendicular to the tangent.
3
Calculate the centre distance.
$\begin{aligned} O P^{2} & = O A^{2} + A P^{2} \\ = 8^{2} + 15^{2} = 289 \\ O P & = 17 cm \end{aligned}$
OP is the hypotenuse of right triangle OAP; select the positive square root.

Answer

PB $= 15$ cm and OP $= 17$ cm.

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$x = 4$ .

Equal tangent segments give $3 x + 1 = 5 x - 7$ , hence $2 x = 8$ and $x = 4$ . Substitution gives a positive length of $13$ for each tangent.

Keep practising

6 worksheet questions · 14 marks

Open worksheet 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 $A B$ make a $52^{\circ}$ angle. $C$ is on the alternate segment. Find angle $A C B$ .
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 $P T$ is 24 cm long and $O P$ is 25 cm, where $O$ is the circle centre. Find the radius $O T$ .
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 $A B$ make a $52^{\circ}$ angle. $C$ is on the alternate segment. Find angle $A C B$ .
2 marks
Apply the alternate-segment theorem to chord $A B$ .
$52^{\circ}$ .
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 $P T$ is 24 cm long and $O P$ is 25 cm, where $O$ is the circle centre. Find the radius $O T$ .
3 marks
$O T$ is perpendicular to $P T$ .
$O T^{2} = 25^{2} - 24^{2} = 49$ , 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 ↗

In this lesson

Understand the idea.

Chords connect length to distance

Tangents have two useful angle facts

See the method in action.

A chord from its centre distance

Define the half-chord.

Find the half-length using Pythagoras.

Recover the full length.

Two tangents

Use equal tangents from P.

Identify the right triangle.

Calculate the centre distance.

Try it yourself.

Check your understanding

Keep practising