AQA Level 2 Further Maths · Level 2 Certificate · Chapter 6: Geometry I · Lesson 2

Circle theorems

About 75 minutesNo calculator57 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/further-maths-circle-theorems/

Date

Do Now

The first is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
An isosceles triangle has one angle of $3 8^{\circ}$ between its two equal sides. Work out the size of each of the other two angles.
[2 marks]
Answer
A2
Last lesson
The angles of a quadrilateral are $(x + 20)^{\circ}$ , $2 x^{\circ}$ , $(3 x - 10)^{\circ}$ and $(2 x + 30)^{\circ}$ . Work out the value of $x$ .
[2 marks]
Answer
A3
From chapter 5
A circle has equation $(x - 3)^{2} + (y + 5)^{2} = 49$ . Write down the coordinates of its centre and its radius.
[2 marks]
Answer
A4
From chapter 4
Solve the simultaneous equations $2 x + y = 100$ and $x + 3 y = 110$ .
[2 marks]
Answer

Example, then your turn

Level 2

Your teacher works through each example with you. Then try the one beside it. $O$ is always the centre.

Example 1

$O$ is the centre. Angle $A O B = 11 2^{\circ}$ . Angle $A C B = x$ and angle $A D B = y$ . Work out $x$ and $y$ . Give reasons.

Not drawn accurately

Hint

x

and the

11 2^{\circ}

stand on the same arc

A B

A C B D

is a cyclic quadrilateral.

B1
$O$ is the centre. Angle $A C B = 4 7^{\circ}$ . Reflex angle $A O B = x$ and angle $A D B = y$ . Work out $x$ and $y$ .
[3 marks]
Not drawn accurately
Answer $x = \dots \dots^{\circ} y = \dots \dots^{\circ}$

Example 2

$A C$ is a diameter. Angle $B A C = 3 1^{\circ}$ . Angle $A C B = x$ and angle $A D B = y$ . Work out $x$ and $y$ . Give reasons.

Not drawn accurately

Hint Angle

A B C

is in a semicircle. Then

x

and

y

stand on the same arc

A B

B2
$A C$ is a diameter. Angle $A D B = 3 8^{\circ}$ and angle $B A C = x$ . Work out $x$ .
[3 marks]
Not drawn accurately
Answer $x = \dots \dots^{\circ}$

Example 3

$P T Q$ is a tangent to the circle at $T$ . Work out the three angles of triangle $A T B$ . Give reasons.

Not drawn accurately

Hint The angle between the tangent and the chord

T A

equals the angle in the other segment, at

B

B3
$P T Q$ is a tangent to the circle at $T$ , and $T A = A B$ . Work out angle $x$ . Give reasons.
[3 marks]
Not drawn accurately
Answer $x = \dots \dots^{\circ}$

Practice

Level 2

C1
$T A$ and $T B$ are tangents to the circle at $A$ and $B$ . Angle $A T B = 4 8^{\circ}$ . Work out angle $A O B$ and angle $A C B$ .
[3 marks]
Not drawn accurately
Answer $A O B = \dots \dots^{\circ} A C B = \dots \dots^{\circ}$
C2
Angle $A C B = (x + 15)^{\circ}$ and reflex angle $A O B = (6 x + 10)^{\circ}$ . Work out the value of $x$ .
[3 marks]
Not drawn accurately
Answer $x = \dots\dots$

C3
$A$ , $B$ , $C$ and $D$ are points on the circle. Angle $B A C = 2 x^{\circ}$ , angle $B D C = (x + 24)^{\circ}$ , angle $A B D = 3 y^{\circ}$ and angle $A C D = (y + 40)^{\circ}$ . Work out the values of $x$ and $y$ .
[4 marks]
Not drawn accurately
Answer $x = \dots\dots y = \dots\dots$
C4
$T Q$ is a tangent to the circle at $T$ . The angle between the tangent and the chord $T A$ is $6 4^{\circ}$ . Work out angle $T O A$ . Give reasons.
[3 marks]
Not drawn accurately
Answer $\dots \dots^{\circ}$
C5
$A B C D$ is a cyclic quadrilateral. $A B E$ is a straight line. Angle $C B E = (x + 25)^{\circ}$ and angle $A D C = (3 x - 15)^{\circ}$ . Work out the size of angle $A B C$ .
[3 marks]
Not drawn accurately
Answer $\dots \dots^{\circ}$
C6
The circle has radius $17$ cm. The chord $A B$ is $30$ cm long. $M$ is on $A B$ and $O M$ is perpendicular to $A B$ . Work out the length $O M$ .
[2 marks]
Not drawn accurately
Answer $\dots\dots$ cm

Exam-style questions

AQA exam style

D1
$A B C D$ is a cyclic quadrilateral with angle $A B C = 7 0^{\circ}$ . Tom says, "Angle $A D C = 14 0^{\circ}$ , because the angle at $D$ is twice the angle at $B$ ."
Tom is wrong. Work out the correct size of angle $A D C$ and give the correct reason.
[2 marks]
Not drawn accurately
Answer $\dots \dots^{\circ}$

D2
$P T Q$ is a tangent to the circle at $T$ . $A B$ is parallel to $P Q$ . Angle $A T P = 5 2^{\circ}$ . $C$ is a point on the circle as shown.
Not drawn accurately
(a)
Work out angle $T B A$ . Give a reason.
[2 marks]
Answer $\dots \dots^{\circ}$
(b)
Show that triangle $T A B$ is isosceles.
[2 marks]
(c)
Hence work out angle $A C B$ . Give a reason.
[2 marks]
Answer $\dots \dots^{\circ}$
Total for D2: 6 marks
D3
$A B$ is a diameter. The tangent at $B$ meets $A C$ extended at $T$ . Angle $C A B = 3 6^{\circ}$ . Work out angle $C B T$ and angle $B T C$ . Give reasons.
[4 marks]
Not drawn accurately
Answer $C B T = \dots \dots^{\circ} B T C = \dots \dots^{\circ}$
D4
$B D$ is a diameter. Angle $B A C = 3 x^{\circ}$ , angle $C A D = (2 x - 10)^{\circ}$ and angle $A B D = (x + 30)^{\circ}$ .
Not drawn accurately
(a)
Work out the value of $x$ .
[3 marks]
Answer $x = \dots\dots$
(b)
Work out angle $A C B$ . Give reasons.
[2 marks]
Answer $\dots \dots^{\circ}$
Total for D4: 5 marks
D5
$O$ is the centre of a circle. $A$ , $B$ and $C$ are points on the circle, with $C$ on the major arc $A B$ . Angle $A O B = 2 y$ . Which of these is angle $A C B$ ?
Circle your answer. $\frac{1}{2} y y 2 y 180 - y$
[1 mark]

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
$O$ is the centre. The chords $A C$ and $B D$ cross at $X$ . Angle $A O B = 10 0^{\circ}$ and angle $C O D = 6 0^{\circ}$ . Work out the size of angle $A X B$ . Give reasons.
[4 marks]
Not drawn accurately
Hint 1 · What to try
Find two angles of triangle $X B C$ using angles at the circumference.
Hint 2 · The first line
Angle $A C B = 5 0^{\circ}$ (half of $A O B$ ) and angle $D B C = 3 0^{\circ}$ (half of $C O D$ ).
Hint 3 · The full method
Angle $A X B$ is the exterior angle of triangle $X B C$ , so it is $50 + 30$ .
Answer $\dots \dots^{\circ}$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use the angle at the centre, semicircle and same segment theorems.
I can use cyclic quadrilaterals and the alternate segment theorem.
I can use tangents to a circle and give a reason for each step.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers