1

The diagram shows a circle, centre O, radius 11 cm.

C, F, G and H are points on the circumference of the circle.
The line AD touches the circle at C and is parallel to the line EG.
B is a point on AD and angle ABO = 140°.

(a) Write down the mathematical name of the straight line AD.

................................................... [1]

(b) (i) Find, in terms of , the circumference of the circle.

................................................... cm [2]
(ii) Work out angle FOH.

Angle FOH = ................................................... [2]

(iii) Calculate the length of the minor arc FH.

................................................... cm [2]
 2

(c) (i) Give a reason why angle BCO is 90°.

...................................................................................................................................... [1]
(ii) Show that BC = 13.11 cm, correct to 2 decimal places.

[3]

(iii) Calculate BH.

BH = ................................................... cm [3]

[Total: 14]

2 The diagram shows four polygons on a 1 cm2 grid.

 3

y
11

10

9
A
8

4
C
3

–7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 x

–1

–2

–3

–4
B
–5

–6

–7

–8

–9

(a) Write down the mathematical name of the shaded polygon.

................................................... [1]
 4

(b) Find the area of the shaded polygon.

................................................... cm2 [2]

(c) Describe fully the single transformation that maps

(i) the shaded polygon onto polygon A,

................................................................................................................................................

................................................................................................................................................ [2]

(ii) the shaded polygon onto polygon B,

................................................................................................................................................

................................................................................................................................................ [3]

(iii) the shaded polygon onto polygon C,

................................................................................................................................................

................................................................................................................................................ [3]

(d) On the grid, draw the image of the shaded polygon after a reflection in the line y = 0. [2]

[Total: 13]

3 The diagram shows the net of a cuboid.

 5

(a) Work out the area of the shaded rectangle, A.

................................................... cm2 [2]

(b) The volume of the cuboid is 468 cm3 .

Complete the statement.

The dimensions of the cuboid are .................... cm by .................... cm by .................... cm [2]

[Total: 4]
 6

A, B and C are points on the circumference of a circle, centre O.

Write down the mathematical name for

(a) the straight line AC,

................................................... [1]

(b) the straight line AB.

................................................... [1]

[Total: 2]

5 The grid shows a point A.

y
4

2
A
1

–4 –3 –2 –1 0 1 2 3 4 x
–1

–2

–3

–4
 7

(a) Write down the coordinates of point A.

( .................... , .................... ) [1]

(b) On the grid, plot the point B at (−1, 3). [1]

(c) C is a point on the grid whose coordinates are whole numbers.

On the grid, mark a point C so that triangle ABC is isosceles. [1]

[Total: 3]

6 The scale drawing shows the positions of three towns A, B and C on a map.
The scale of the map is 1 centimetre represents 10 kilometres.

North

North

Scale 1 cm to 10 km
B

(a) Find the actual distance AB.

Answer(a) .........................................km [1]

(b) Measure the bearing of A from B.

Answer(b) ......................................... [1]

(c) Write the scale 1 cm to 10 km in the form 1 : n.

Answer(c) 1 : .............................. [1]

[Total: 3]
 8

The diagram shows a quadrilateral PQRS which is made from four congruent triangles A, B, C and D.

(a) Write down the mathematical name for the quadrilateral PQRS.

................................................... [1]

(b) (i) Write down the co-ordinates of S.

( .................... , .................... ) [1]

(ii) Measure the obtuse angle PSR.

................................................... [1]

(c) (i) Measure the length of the line PQ.

................................................... cm [1]
(ii) Work out the perimeter of the quadrilateral PQRS.

................................................... cm [1]

[Total: 5]
 9

8 Using a straight edge and compasses only, construct the equilateral triangle ABC.
The base AB has been drawn for you.

[2]

[Total: 2]

9 The diagram shows the net of a triangular prism on a 1 cm2 grid.

 10

(a) Write down the mathematical name for the type of triangle shown on the grid.

................................................... [1]

(b) (i) Measure the perpendicular height of the triangle.

................................................... cm [1]
 11

(ii) Calculate the area of the triangle.

................................................... cm2 [2]

(iii) Calculate the volume of the triangular prism.

................................................... cm3 [2]

[Total: 6]

10 Using a straight edge and compasses only, construct the equilateral triangle ABC.
Side AB has been drawn for you.

[2]

[Total: 2]

11 The scale drawing shows the positions of Annika’s house, A, and Bernhard’s house, B, on a map.
The scale is 1 centimetre represents 300 metres.
 12

(a) Measure the bearing of Bernhard’s house from Annika’s house.

................................................... [1]
 13

(b) Dougie’s house is

• on a bearing of 320° from Bernhard’s house

and
• 1650 m from Annika’s house.

Mark on the map the two possible positions of Dougie’s house.

Label each of these points D.
[4]

[Total: 5]

12 The scale drawing shows two boundaries, AB and BC, of a field ABCD.
The scale of the drawing is 1 cm represents 8 m.

The boundaries CD and AD of the field are each 72 m long.

(a) Work out the length of CD and AD on the scale drawing.

................................................... cm [1]
 14

(b) Using a ruler and compasses only, complete accurately the scale drawing of the field. [2]

[Total: 3]

13 The scale drawing shows town A, town B and town C on a map.

There is a straight road between town A and town B.
The scale of the map is 1 centimetre represents 8 kilometres.

(a) Measure the bearing of town A from town B.

................................................... [1]

(b) Write the scale of the map in the form 1 : n.

1 : ................................................... [1]
 15

(c) A straight road from town C is on a bearing of 246°.

It meets the road from town A to town B at point X.

On the map, draw the road from town C to point X.

Label the position of X.
[1]

(d) Josie is at point X at 10 50.

She arrives at town B 37 minutes later.

Work out the time that she arrives at town B.

................................................... [1]

[Total: 4]

14 Point B is 36 km from point A on a bearing of 140°.

 16

(a) Using a scale of 1 centimetre to represent 4 kilometres, mark the position of B.

Scale: 1 cm to 4 km [2]

(b) (i) Point C is 28 km from A and 20 km from B.

The bearing of C from A is less than 140°.

Using a ruler and compasses only, construct triangle ABC.

Show all your construction arcs.
[3]
(ii) Measure angle ACB.

Angle ACB = ................................................... [1]

[Total: 6]

15 A solid hemisphere has volume 230 cm3.

 17

(a) Calculate the radius of the hemisphere.

[The volume, V, of a sphere with radius r is .]

................................................... cm [3]

(b) A solid cylinder with radius 1.6 cm is attached to the hemisphere to make a toy.

The total volume of the toy is 300 cm3.

(i) Calculate the height of the cylinder.

................................................... cm [3]
 18

(ii) A mathematically similar toy has volume 19 200 cm3.

Calculate the radius of the cylinder for this toy.

................................................... cm [3]

[Total: 9]

16 Marianne sells two sizes of photo.

These photos are mathematically similar rectangles.

The smaller photo has length 15 cm and width 12 cm.

The larger photo has area 352.8 cm2.

Calculate the length of the larger photo.

................................................... cm [3]

[Total: 3]
 19

17

In the diagram, AB and CD are parallel.

AD and BC intersect at right angles at the point X.
AB = 10 cm, CD = 5 cm, AX = 8 cm and BX = 6 cm.

(a) Use similar triangles to calculate DX.

DX = ................................................... cm [2]

(b) Calculate angle XAB.

Angle XAB = ................................................... [2]

[Total: 4]
 20

18 These two flags are mathematically similar.

Calculate the height, h, of the second flag.

h = ................................................... m [2]

[Total: 2]

19

Triangle ABC is mathematically similar to triangle PQR.

The area of triangle ABC is 16 cm2.

(a) Calculate the area of triangle PQR.

................................................... cm2 [2]

 21

(b) The triangles are the cross-sections of prisms which are also mathematically similar.
The volume of the smaller prism is 320 cm3.

Calculate the length of the larger prism.

................................................... cm [3]

[Total: 5]

20

D
B

63.4° 63.4° E

Two new triangles, D and E, are made from triangle B, as shown in the diagram.

Are all three triangles similar?

Give a reason for your answer.

........................ because ...........................................................................................................................

.................................................................................................................................................................... [2]

[Total: 2]
 22

21 A quadrilateral P is enlarged by a scale factor of 1.2 to give quadrilateral Q.

The area of quadrilateral P is 20 cm2.

Calculate the area of quadrilateral Q.

................................................... cm2 [2]

[Total: 2]

22 Draw all the lines of symmetry on each shape.

[4]

[Total: 4]

23 (a) Complete the table of values for .

x −3 −2 −1 0 1 2 3 4

y 7 −3 −5

[3]
 23

23 (b) On the grid, draw the graph of for .

[4]

(c) Write down the co-ordinates of the lowest point on the graph.

( .................... , .................... ) [1]

(d) (i) On the grid, draw the line of symmetry of the graph. [1]
(ii) Write down the equation of this line.

................................................... [1]

[Total: 10]
 24

24

The diagram shows a cuboid.

AB = 8 cm, BC = 4 cm and CR = 5 cm.

(a) Write down the number of planes of symmetry of this cuboid.

................................................... [1]

(b) Calculate the angle between the diagonal AR and the plane BCRQ.

................................................... [4]

[Total: 5]
 25

25 Draw all the lines of symmetry on the rectangle below.

[2]

[Total: 2]

26 Soraya makes rectangular flags.

(a) On the rectangle, draw the lines of symmetry. [2]

(b) Each flag measures 1.2 m by 1.8 m.

Calculate the area of one flag.

................................................... m2 [2]

[Total: 4]

27 The diagram shows a regular polygon.

(a) Write down the mathematical name for this shape.

................................................... [1]
 26

(b) Write down the order of rotational symmetry of this shape.

................................................... [1]

[Total: 2]

28

The diagram shows a rhombus.

(a) Write down the order of rotational symmetry.

................................................... [1]

(b) On the diagram, draw all the lines of symmetry. [2]

[Total: 3]

29

The diagram shows a circle, centre F and diameter BG.

AC is a tangent to the circle at B.
BF is parallel to DE, angle GFE = 72° and angle BCD = angle CDE.

(a) Write down the mathematical name of the polygon BCDEF.

................................................... [1]
 27

(b) Explain why angle FBC is a right angle.

.......................................................................................................................................................... [1]

(c) Find angle BFE, giving a reason for your answer.

Angle BFE = .............................. because ........................................................................................

.......................................................................................................................................................... [2]

(d) Find angle FED.

Angle FED = ................................................... [1]

(e) Calculate angle BCD.

Angle BCD = ................................................... [4]

[Total: 9]

30

A, B and C are points on the circumference of a circle, centre O.

(a) Give a geometrical reason why angle ABC = 90°.

.......................................................................................................................................................... [1]

(b) AB = 20 cm and AC = 52 cm.

 28

(i) Use trigonometry to calculate angle BAC.

Angle BAC = ................................................... [2]

(ii) Show that BC = 48 cm.

[2]

(iii) Work out the area of triangle ABC.

................................................... cm2 [2]

(iv) Work out the total shaded area.

................................................... cm2 [3]

[Total: 10]
 29

31 The diagram shows an isosceles triangle and a straight line AB.

Find the value of x and the value of y.

x = ...................................................

y = ................................................... [2]

[Total: 2]

32 Find the size of one interior angle of a regular decagon.

................................................... [3]

[Total: 3]
 30

33

In the diagram, AB = AC.

Find

(a) angle BAC,

Angle BAC = ................................................... [1]

(b) angle ABC.

Angle ABC = ................................................... [1]

[Total: 2]
 31

34 The exterior angle of a regular polygon is x° and the interior angle is 8x°.

Calculate the number of sides of the polygon.

................................................... [3]

[Total: 3]

35 In a regular polygon, the interior angle is 11 times the exterior angle.

(a) Work out the number of sides of this polygon.

................................................... [3]

(b) Find the sum of the interior angles of this polygon.

................................................... [2]

[Total: 5]
 32

36 The diagram shows a right-angled triangle.

Find the value of x.

x = ................................................... [1]

[Total: 1]

37

The diagram shows an isosceles triangle, ABC.

BCD is a straight line.

Find the value of x.

x = ................................................... [2]

[Total: 2]
 33

38

P, Q, R, S and T lie on the circle, centre O.

Angle PST = 75° and angle QTS = 85°.

Find the values of v, w, x and y.

v = ...................................................

w = ...................................................

x = ...................................................

y = ................................................... [6]

[Total: 6]
 34

39

In the diagram, A, B, C and D lie on the circle, centre O.

EA is a tangent to the circle at A.
Angle EAB = 61° and angle BAC = 55°.

(a) Find angle BAO.

Angle BAO = ................................................... [1]

(b) Find angle AOC.

Angle AOC = ................................................... [2]

(c) Find angle ABC.

Angle ABC = ................................................... [1]

(d) Find angle CDA.

Angle CDA = ................................................... [1]

[Total: 5]
 35

40 Each interior angle of a regular polygon is 162°.

Calculate the number of sides of the polygon.

................................................... [3]

[Total: 3]