Sec 3 · G3 (Express)

Circle properties

Practise circle properties with step-by-step help. Free account.

1. The circle properties

Angle at centre = 2 × angle at circumference

\displaystyle 120^\circ = 2 \times 60^\circ

Angle AOB at the centre is 120°, twice angle ACB at the circumference, 60°.

Reason: $\angle$ at centre $= 2\angle$ at circumference

Angle in a semicircle = 90°

\displaystyle \angle ACB = 90^\circ

AB is a diameter; angle ACB at C is a right angle.

Reason: $\angle$ in semicircle

Angles in the same segment are equal

\displaystyle \angle ACB = \angle ADB

Angles ACB and ADB stand on the same arc AB and are both 70°.

Reason: $\angle$ s in same segment

Opposite angles of a cyclic quadrilateral add up to 180°

\displaystyle 95^\circ + 85^\circ = 180^\circ

Cyclic quadrilateral ABCD with angle A 95° and angle C 85°.

Reason: $\angle$ s in opp. segments

Exterior angle = interior opposite angle

\displaystyle \angle DCE = \angle DAB

BC produced to E; angle DCE is 95°, equal to angle DAB, 95°.

Reason: ext. $\angle$ of cyclic quad.

Tangent meets the radius at 90°

\displaystyle \angle OAT = 90^\circ

Tangent AT at A, with a right angle between radius OA and the tangent.

Reason: tan. $\perp$ rad.

Tangents from an external point are equal

\displaystyle PA = PB

Tangents PA and PB from P, marked equal, each at right angles to its radius; OP cuts the angle at P into two equal halves and the angle at O into two equal halves.

Reason: tangents from ext. pt.
$OP$ bisects $\angle APB$ and $\angle AOB$

The perpendicular from the centre bisects the chord

\displaystyle AM = MB

OM perpendicular to chord AB at M, with AM and MB marked equal.

Reason: $\perp$ from centre bisects chord

Equal chords are equidistant from the centre

\displaystyle AB = CD \iff OM = ON

Chords AB and CD of equal length, all four half-chords marked equal; OM and ON are perpendicular to them, marked equal.

Reason: equal chords equidistant from centre
and the other way: chords equidistant from centre are equal

To show	Show that
$C$ lies on the circle with diameter $AB$	$\angle ACB = 90^\circ$ (or $AC^2 + BC^2 = AB^2$ )
$A$ , $B$ , $C$ , $D$ lie on a circle	a pair of opposite angles of $ABCD$ adds up to $180^\circ$
$AP$ is a tangent at $A$	$\angle OAP = 90^\circ$ (or $OA^2 + AP^2 = OP^2$ )

2. Angle at the centre

📖 Worked example

In the diagram, $A$ , $B$ and $C$ lie on the circle with centre $O$ . Find $x$ .

Step 1 of 1

Find ∠AOB

\mathstrut \angle AOB = 2 \times 56^\circ

(∠ at centre = 2 ∠ at circumference)

x = 112

📖 Worked example

In the diagram, $A$ , $B$ and $C$ lie on the circle with centre $O$ . Find $x$ .

Step 1 of 2

Which angle at O stands on the same arc as the angle at C?

\mathstrut \text{Reflex } \angle AOB = 360^\circ - 154^\circ

(∠s at a point)

\text{Reflex } \angle AOB = 206^\circ

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3. Angle in a semicircle

📖 Worked example

In the diagram, $AB$ is a diameter of the circle with centre $O$ , and $C$ lies on the circle. Find $x$ .

Step 1 of 3

Find ∠OBC

\angle OBC = 56^\circ

(base ∠s of isos. △)

📖 Worked example

In triangle $ABC$ , $AC = 9$ cm, $BC = 12$ cm and $AB = 15$ cm. A circle is drawn with $AB$ as its diameter. Does the circle pass through $C$ ? Choose the answer with its reason.

Step 1 of 3

Compare AC² + BC² with AB²

AC^2 + BC^2 = 9^2 + 12^2

\phantom{AC^2 + BC^2} = 225

AB^2 = 15^2

\phantom{AB^2} = 225

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4. Angles in the same segment

📖 Worked example

In the diagram, $A$ , $B$ , $C$ and $D$ lie on the circle. The chords $AC$ and $BD$ meet at $X$ . Find $x$ .

Step 1 of 3

Find ∠ACD

\angle ACD = \angle ABD

(∠s in same segment)

\angle ACD = 26^\circ

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5. Cyclic quadrilaterals

📖 Worked example

In the diagram, $A$ , $B$ , $C$ and $D$ lie on the circle and $BC$ is produced to $E$ . Find $x$ .

Step 1 of 2

Find ∠BCD

\mathstrut \angle BCD = 180^\circ - 88^\circ

(∠s in opp. segments)

\angle BCD = 92^\circ

📖 Worked example

The diagram shows a quadrilateral $ABCD$ . Do $A$ , $B$ , $C$ and $D$ lie on a circle? Choose the answer with its reason.

Step 1 of 3

Find angle BCD (angle sum of quadrilateral)

\angle BCD = 360^\circ - 79^\circ - 93^\circ - 92^\circ

\phantom{\angle BCD} = 96^\circ

(∠ sum of quad.)

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6. Tangent and radius

📖 Worked example

In the diagram, $PA$ is a tangent to the circle with centre $O$ at $A$ . Find $x$ , correct to 3 significant figures.

Step 1 of 2

Find ∠OAP

\angle OAP = 90^\circ

(tan. ⊥ rad.)

📖 Worked example

In the diagram, $AT$ is a tangent to the circle with centre $O$ at $A$ , and $AB$ is a chord. Find $x$ .

Step 1 of 2

Find ∠OAT, then ∠OAB

\angle OAT = 90^\circ

(tan. ⊥ rad.)

\mathstrut \angle OAB = 90^\circ - 24^\circ

\angle OAB = 66^\circ

📖 Worked example

$A$ is a point on the circle with centre $O$ and radius $9$ cm. Is $AP$ a tangent to the circle at $A$ ? Choose the answer with its reason.

Step 1 of 3

Compare OA² + AP² with OP²

OA^2 + AP^2 = 9^2 + 12^2

\phantom{OA^2 + AP^2} = 225

OP^2 = 15^2

\phantom{OP^2} = 225

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7. Tangents from an external point

📖 Worked example

In the diagram, $PA$ and $PB$ are tangents to the circle with centre $O$ . Find $x$ .

Step 1 of 2

Compare PA and PB

PA = PB

(tangents from ext. pt.)So triangle PAB is isosceles.

📖 Worked example

In the diagram, $PA$ and $PB$ are tangents to the circle with centre $O$ , and $OP$ is joined. Find $x$ .

Step 1 of 3

Find ∠OPA

\angle OPA = \angle OPB

(tangents from ext. pt.)

\angle OPA = 34^\circ

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8. Perpendicular bisector of a chord

📖 Worked example

In the diagram, $AB$ is a chord of the circle with centre $O$ , and $OM$ is perpendicular to $AB$ . Find the length of the chord $AB$ , correct to 3 significant figures.

Step 1 of 2

Pythagoras' theorem in triangle OAM

AM^2 = OA^2 - OM^2

\mathstrut AM^2 = 10^2 - 7^2

AM^2 = 51

AM = 7.1414\ldots

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9. Equal chords

📖 Worked example

In the diagram, $AB$ and $CD$ are chords of the circle with centre $O$ . $OM$ is perpendicular to $AB$ and $ON$ is perpendicular to $CD$ . Find the length of the chord $CD$ .

Step 1 of 4

Compare the chords AB and CD

AB = CD

(chords equidistant from centre are equal)So finding

CD

gives

AB

too.

📖 Worked example

In the diagram, $AB$ and $CD$ are chords of the circle with centre $O$ . $OM$ is perpendicular to $AB$ and $ON$ is perpendicular to $CD$ . Both chords are $16$ cm long. Find $x$ .

Step 1 of 3

Find OM

OM = ON

(equal chords are equidistant from centre)

OM = 6

Practise this →

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Sec 3 · G3 (Express)

Circle properties

Practise circle properties with step-by-step help. Free account.

Start free Worksheet (PDF)

1. The circle properties

Angle at centre = 2 × angle at circumference

\displaystyle 120^\circ = 2 \times 60^\circ

Reason: $\angle$ at centre $= 2\angle$ at circumference

Angle in a semicircle = 90°

\displaystyle \angle ACB = 90^\circ

Reason: $\angle$ in semicircle

Angles in the same segment are equal

\displaystyle \angle ACB = \angle ADB

Reason: $\angle$ s in same segment

Opposite angles of a cyclic quadrilateral add up to 180°

\displaystyle 95^\circ + 85^\circ = 180^\circ

Reason: $\angle$ s in opp. segments

Exterior angle = interior opposite angle

\displaystyle \angle DCE = \angle DAB

Reason: ext. $\angle$ of cyclic quad.

Tangent meets the radius at 90°

\displaystyle \angle OAT = 90^\circ

Reason: tan. $\perp$ rad.

Tangents from an external point are equal

\displaystyle PA = PB

Reason: tangents from ext. pt.
$OP$ bisects $\angle APB$ and $\angle AOB$

The perpendicular from the centre bisects the chord

\displaystyle AM = MB

Reason: $\perp$ from centre bisects chord

Equal chords are equidistant from the centre

\displaystyle AB = CD \iff OM = ON

Reason: equal chords equidistant from centre
and the other way: chords equidistant from centre are equal

To show	Show that
$C$ lies on the circle with diameter $AB$	$\angle ACB = 90^\circ$ (or $AC^2 + BC^2 = AB^2$ )
$A$ , $B$ , $C$ , $D$ lie on a circle	a pair of opposite angles of $ABCD$ adds up to $180^\circ$
$AP$ is a tangent at $A$	$\angle OAP = 90^\circ$ (or $OA^2 + AP^2 = OP^2$ )

2. Angle at the centre

📖 Worked example

In the diagram, $A$ , $B$ and $C$ lie on the circle with centre $O$ . Find $x$ .

Step 1 of 1

Find ∠AOB

\mathstrut \angle AOB = 2 \times 56^\circ

(∠ at centre = 2 ∠ at circumference)

x = 112

📖 Worked example

In the diagram, $A$ , $B$ and $C$ lie on the circle with centre $O$ . Find $x$ .

Step 1 of 2

Which angle at O stands on the same arc as the angle at C?

\mathstrut \text{Reflex } \angle AOB = 360^\circ - 154^\circ

(∠s at a point)

\text{Reflex } \angle AOB = 206^\circ

Practise this →

3. Angle in a semicircle

📖 Worked example

In the diagram, $AB$ is a diameter of the circle with centre $O$ , and $C$ lies on the circle. Find $x$ .

Step 1 of 3

Find ∠OBC

\angle OBC = 56^\circ

(base ∠s of isos. △)

📖 Worked example

In triangle $ABC$ , $AC = 9$ cm, $BC = 12$ cm and $AB = 15$ cm. A circle is drawn with $AB$ as its diameter. Does the circle pass through $C$ ? Choose the answer with its reason.

Step 1 of 3

Compare AC² + BC² with AB²

AC^2 + BC^2 = 9^2 + 12^2

\phantom{AC^2 + BC^2} = 225

AB^2 = 15^2

\phantom{AB^2} = 225

Practise this →

4. Angles in the same segment

📖 Worked example

In the diagram, $A$ , $B$ , $C$ and $D$ lie on the circle. The chords $AC$ and $BD$ meet at $X$ . Find $x$ .

Step 1 of 3

Find ∠ACD

\angle ACD = \angle ABD

(∠s in same segment)

\angle ACD = 26^\circ

Practise this →

5. Cyclic quadrilaterals

📖 Worked example

In the diagram, $A$ , $B$ , $C$ and $D$ lie on the circle and $BC$ is produced to $E$ . Find $x$ .

Step 1 of 2

Find ∠BCD

\mathstrut \angle BCD = 180^\circ - 88^\circ

(∠s in opp. segments)

\angle BCD = 92^\circ

📖 Worked example

The diagram shows a quadrilateral $ABCD$ . Do $A$ , $B$ , $C$ and $D$ lie on a circle? Choose the answer with its reason.

Step 1 of 3

Find angle BCD (angle sum of quadrilateral)

\angle BCD = 360^\circ - 79^\circ - 93^\circ - 92^\circ

\phantom{\angle BCD} = 96^\circ

(∠ sum of quad.)

Practise this →

6. Tangent and radius

📖 Worked example

In the diagram, $PA$ is a tangent to the circle with centre $O$ at $A$ . Find $x$ , correct to 3 significant figures.

Step 1 of 2

Find ∠OAP

\angle OAP = 90^\circ

(tan. ⊥ rad.)

📖 Worked example

In the diagram, $AT$ is a tangent to the circle with centre $O$ at $A$ , and $AB$ is a chord. Find $x$ .

Step 1 of 2

Find ∠OAT, then ∠OAB

\angle OAT = 90^\circ

(tan. ⊥ rad.)

\mathstrut \angle OAB = 90^\circ - 24^\circ

\angle OAB = 66^\circ

📖 Worked example

$A$ is a point on the circle with centre $O$ and radius $9$ cm. Is $AP$ a tangent to the circle at $A$ ? Choose the answer with its reason.

Step 1 of 3

Compare OA² + AP² with OP²

OA^2 + AP^2 = 9^2 + 12^2

\phantom{OA^2 + AP^2} = 225

OP^2 = 15^2

\phantom{OP^2} = 225

Practise this →

7. Tangents from an external point

📖 Worked example

In the diagram, $PA$ and $PB$ are tangents to the circle with centre $O$ . Find $x$ .

Step 1 of 2

Compare PA and PB

PA = PB

(tangents from ext. pt.)So triangle PAB is isosceles.

📖 Worked example

In the diagram, $PA$ and $PB$ are tangents to the circle with centre $O$ , and $OP$ is joined. Find $x$ .

Step 1 of 3

Find ∠OPA

\angle OPA = \angle OPB

(tangents from ext. pt.)

\angle OPA = 34^\circ

Practise this →

8. Perpendicular bisector of a chord

📖 Worked example

In the diagram, $AB$ is a chord of the circle with centre $O$ , and $OM$ is perpendicular to $AB$ . Find the length of the chord $AB$ , correct to 3 significant figures.

Step 1 of 2

Pythagoras' theorem in triangle OAM

AM^2 = OA^2 - OM^2

\mathstrut AM^2 = 10^2 - 7^2

AM^2 = 51

AM = 7.1414\ldots

Practise this →

9. Equal chords

📖 Worked example

In the diagram, $AB$ and $CD$ are chords of the circle with centre $O$ . $OM$ is perpendicular to $AB$ and $ON$ is perpendicular to $CD$ . Find the length of the chord $CD$ .

Step 1 of 4

Compare the chords AB and CD

AB = CD

(chords equidistant from centre are equal)So finding

CD

gives

AB

too.

📖 Worked example

In the diagram, $AB$ and $CD$ are chords of the circle with centre $O$ . $OM$ is perpendicular to $AB$ and $ON$ is perpendicular to $CD$ . Both chords are $16$ cm long. Find $x$ .

Step 1 of 3

Find OM

OM = ON

(equal chords are equidistant from centre)

OM = 6

Practise this →

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