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THOMAS WHITHAM SIXTH FORM

Unit 3

Mathematics Geometry GCSE

S J Cooper

t h o m a s w h i t h a m . p b w o r k s . c o m

S J Cooper

Geometry (3) Constructions of triangles with protractor

REMEMBER DO NOT REMOVE ANY CONSTRUCTION LINES OR ARCS.

1. Draw a triangle ABC whose sides are AB = 8cm, AC = 6cm and BC = 4cm.

Measure and write down the size of CBA ˆ .

2. Draw a triangle LMN where 6LM cm, 35ˆ NLM and 7LN cm.

Measure and write down the size of NML ˆ .

3. Draw the triangle XYZ when 5.3XY cm, 8.5YZ cm and 68ˆ ZYX

Measure and write down the length of XZ.

4. Draw a triangle HIG whose sides are HI = 6.4cm, HG = 7.1cm and IG = 4.5cm.

Measure and write down the size of GIHˆ .

5. Draw the triangle STU when 3.8ST cm, 9.4TU cm and 29ˆ STU

Measure and write down the length of ST.

6. Draw accurate diagrams for each of the triangles below and find the lengths required.

(a) (b)

Find Angle E Find length F

(c) (d)

Find Angle N Find Angle M

7cm

10cm 10cm

42

10cm

8cm

E 36

4cm

122

F

N

94

5.4cm

8.1cm

M

S J Cooper

Geometry (4) Constructions of triangles without protractor

REMEMBER DO NOT REMOVE ANY CONSTRUCTION LINES OR ARCS.

7. Draw a triangle ABC whose sides are AB = 8cm, AC = 6cm and angle 60ABC .

Measure and write down the size of CAB ˆ .

8. Draw a triangle DEF where 5DE cm, 7DF cm and 90ˆ FDE

Measure and write down the size of FED ˆ

9. Draw the triangle XYZ when 2.4XY cm, 3.6YZ cm and 30ˆ ZYX

Measure and write down the length of XZ.

10. Draw the triangle STU when 5.7ST cm, 45ˆ UST and 30ˆ STU

Measure and write down the length of ST.

11. Draw the triangle LMN when 7.4LM cm, 45ˆ NML and 60ˆ NLM

Measure and write down the length of MN.

12. Construct each of the following triangles

(a) (b)

(c)

13. (a) Construct triangle ABC where AB = 5cm, BC = 6cm and AC = 4.3cm.

(b) Bisect the side given by the line AB.

(c) Bisect the each of the other lines AC and BC.

(d) Hence using the point of trisection as the centre draw a circle which touches the vertices A,

B and C.

8cm

6cm

7.5cm

30

135 30

5cm

S J Cooper

14. Using ruler and compasses only construct a rectangle with dimensions 7cm by 4cm.

15. Construct the rectangle ABCD where AB = 9cm and BC = 5.3cm

State the length of the diagonal AC.

16. Construct the trapezium below.

60

9cm

6cm

S J Cooper

Geometry (5) Area and perimeter

1. Find the area for each of the following shapes without the use of a calculator.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

2. A rectangle of area 216 m2 has length 18m what is its width?

3. A triangle with base length 15cm has an area of 210cm2, calculate the height of this triangle.

4. A rectangle has a perimeter of 48cm and a length of 7cm. Calculate the width of the rectangle.

5. A room has a rectangular floor with dimensions 7.5m by

6.4m. If a rug with dimensions 4.6m by 3.7m is placed onto

the floor calculate the area of flooring not covered by the

rug.

6. A photograph 27cm by 25cm is placed into a frame with dimensions 34cm by 28cm.

(a) What is the area of the frame?

(b) What is the area of the photograph?

(c) What is the area of the frame visible when the

photograph is in place?

14cm

26cm

23m

23m 23cm

12cm

8cm

5cm

12cm 23mm

21mm

19mm

9cm

13cm

7cm

8cm

7cm

15cm

12cm

10cm

3m

13m

7.5m

6.4m

4.6 m

3.7 m

S J Cooper

Geometry (6) Area & perimeter II

1. Calculate the area for each of the following circles, giving your answers correct to 1 decimal

place.

(a) (b) (c) (d)

2. Calculate the area of each of the following circles giving your answers correct to 2 decimal

places.

(a) (b) (c) (d)

3. Which has the greater area, a circle with radius 9cm or a square with side 15cm?

4. Given the area of a circle is 54cm2 find its radius correct to 2 decimal places.

5. Find the area of the semicircle drawn opposite,

giving your answer to 2 decimal places.

6. Find the area of the shape opposite, giving your

answer correct to 1 decimal place.

7. Calculate the shaded area for each of the following shapes. [giving your answers correct to 2

significant figures]

(a) (b) (c)

8. A circle has an area of 125 cm2. Calculate the length of its radius, giving your answer to 2

decimal places.

5m 14cm 14cm 54m

3.7m 7.2cm 75m 60km

36 cm

100 cm

32 cm

12cm 6cm 5m 12m

7m

S J Cooper

Geometry (7) Circumference of a circle

1. Calculate the circumference of each of the following circles, giving your answers correct to one

decimal place.

(a) (b) (c) (d)

2. Calculate the circumference of each of the following circles, giving your answers correct to 2

decimal places.

(a) (b) (c) (d)

3. Find the perimeter of the semicircle drawn opposite,

giving your answer to 2 decimal places.

4. (a) What is the perimeter of a circle of diameter 70 metres (correct to 2 decimal places)?

The diagram is of a running track with “straights” of

length 150m and with semicircular „bends‟ which have

diameter 70m.

(b) What is the length of one complete lap?

(c) How many laps (approximately) must an athlete run in a race of 10 000m?

5. A bicycle wheel has diameter 75cm. Calculate its circumference, giving your answer correct to

the nearest whole number.

6. What is the diameter of a circle whose circumference is 24cm? [Give your answer correct to 1

decimal place].

7. What is the circumference of a circle whose area is 60cm2? {give your answer correct to the

nearest whole number]

8. Which has the greatest perimeter, a circle with radius 6cm or a square with side 5cm?

7cm 23m 19cm 43m

6.1m 2.3cm 112m 38m

50 cm

150m

70m

S J Cooper

Geometry (8) Area & perimeter of irregular shapes

1. Work out the area and perimeters for each of the following irregular shapes.

(a) (b) (c)

2. Work out the area for each of the following irregular shapes.

(a) (b) (c)

3. Work out the shaded area for each of the following (all measurements are given in centimetres):

(a) (b)

(c) (d)

4cm

6cm

12cm

13cm 23cm

7cm

7cm

23cm 14cm

19m

21m 7m

4m

7m 12m

17m

12m

22m

24cm

38cm

7cm

15m

8m

9m 21m 7m

5

11 21

15

7 7

7

23

13

3

8

S J Cooper

4. Calculate the areas for each of the following shapes.

(a) (b)

(c) (d)

(e) (f)

18

27

18

30

18

18

4

7

15

10 1 1

14

27

25

13

9 9

30

10

7

7

S J Cooper

Geometry (9) Volume of a prism

1. Without a calculator find (a) the base area (b) the volume for the following cuboids

(i) (ii) (iii) (iv)

(v) (vii) (viii)

2. A classroom has a volume of 380m7 , if the length and width of the room are 8m and 7.5m

respectively, how high is this classroom?

3. Bricks with dimensions 25cm by 12cm by 9cm are being used to build a wall.

(a) Find the volume of one brick (i) in 3cm (ii) in 3m .

(b) If the wall is to have a total volume of 675 3m , how many brick will we need ?

4. Without a calculator find the volume for each of the following triangular based prisms.

(a) (b) (c)

(d) (e) (g)

5. Workout the volume for each of the following, giving your answers to 2 decimal places.

19cm

5cm

7cm

5m

14m

9m

8m

11m

7cm

32cm

15cm

8cm

5cm

13cm

11cm

17m

16m

24m

4cm

7cm

5cm

6m

6m

5m

0.5m

11m

4m

3cm

10cm

6cm 7m

5m

12m

11cm

4cm

6cm

8m

12cm

15cm

S J Cooper

(a) (b) (c) (d)

(e) (f) (g) (h)

6. For each of the following calculate

(i) the base area (ii) the volume, given that all measurements are in cm.

(a) (b) (c)

(d) (e)

7. A drain pipe of length 5metres has inner circle with diameter 8cm

and outer diameter with diameter 9cm. Work out the volume of

this drainpipe.

9cm

14cm

14cm

6cm

3m

16m

9cm

13cm

6.5m

7.5m

2cm

7cm

4.2m

3.8m

8.9cm

7.8cm

10

5

7

6

6

9

12

6

7

5

8

7 15

10

19

17

9

8

8

10

20

15

S J Cooper

Geometry (10) Types of Polygons

1. Name each of the following types of triangles

(a) (b) (c)

(d) (e) (f)

2. Draw a set of axes from –6 to 6 for each of the following problems. Plot the coordinates for each

of the following. Join up the points to form the quadrilateral ABCD. What name is given to each

shape drawn?

(i) A(–4, 2) , B(–2, 4) , C(–4, 6) , D(–6, 4)

(ii) A(2, 0) , B(3, –3) , C(4, 0) , D(3, 1)

(iii) A(–3, –3) , B(–3, –6) , C(0, –4) , D(0, –1)

(iv) A(2, 1) , B(6, 1) , C(4, 5) , D(2, 5)

(v) A(–4,1) , B(–4, –2) , C(1, –2) , D(1, 1)

3. (a) What name best describes a rhombus with all angles at 90 ?

(b) What name best describes a parallelogram with all sides equal in length?

(c) What name best describes a parallelogram with all sides equal in length and all angles at

90?

4. Name each of the following quadrilaterals

(a) CJLK

(b) ABCD

(c) FGHI

(d) CDEF

(e) CFIJ

(f) DEIJ

H G

F E

C

B A

D

I

J

K L

S J Cooper

Geometry (11) Solids

1. For each of the tabulated solids below count the number of faces, vertices (corners) and edges.

Enter the numbers in the appropriate place. In the last column work out the value of F + V – E

for each line. State what you notice.

Number of

Faces (F)

Number of

Vertices (V)

Number of

Edges (E)

F + V – E

Cube

Cuboid

Square based pyramid

Tetrahedron

Triangular prism

2. A Sweet is of the shape of a triangular prism until

someone cuts away a corner with a knife, as shown.

Count up faces, vertices and edges on the remainder of

the butter shown. Complete the following.

F =……….. V = …………. E = ………….

F + V – E = …………….

3. (a) Using a pencil draw a sketch of a tetrahedron. Now take away the top corner

using a rubber and redraw it to look as though someone had cut it away.

(b) Complete the following for the remainder of the shape.

F =……….. V = …………. E = ………….

F + V – E = …………….

4. Here are some views of geometrical solids of the type drawn in class. State which they could

be. [Some will have more than one answer!]

(i) (ii) (iii) (iv)

S J Cooper

5. This is a cuboid (edges not equal in length)

Use tracing paper to copy the outline and dotted

(hidden) lines into your exercise book. On your

diagram draw different planes of symmetry.

6. This is a cube (all edges equal). It will have three

planes of symmetry similar to the cuboid in

question 5.

i) Use tracing paper to copy the outline and

dotted lines into your exercise book. Draw a

plane of symmetry.

ii) Repeat the exercise of (i) as many times as

you need to until all planes of symmetry

have been found.

iii) How many planes of symmetry does the

cube have?

7. This is a square based pyramid

i) Use tracing paper to copy the outline and dotted

lines into your exercise book. Draw a plane of

symmetry.

ii) Repeat the exercise of (i) as many times as you

can have until all planes of symmetry have been

found.

8. This is a sphere with a plane of symmetry. Draw a

sphere into your book along with another plane of

symmetry.

How many planes of symmetry could be drawn?

S J Cooper

9. This is a cylinder.

Draw a cylinder into your book with a different plane of symmetry.

How many planes of symmetry could be drawn?

10. (i) Using a square (side 2cm) complete a net for a square

based pyramid each edge of which will be length 6 cm.

(ii) Draw on card a net for a square based pyramid of length

6cm. Add suitable flaps, cut out your net and glue together.

11. Draw an accurate construction for the net of a tetrahedron with edges 5cm in length.

Draw some flaps. Cut out your net; Use a pritt stick to glue together in the form of a regular

tetrahedron.

S J Cooper

Geometry (12) Angles in a straight line

Work out the lettered angles for each of the following diagrams.

Remember to show your working. All diagrams are not drawn to scale.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14. 15.

53 a 144

b 17

c 55

31

d 75 9 e 83

11

f 79 26

3m m 2n

2n n 42 r

2r

42 p

p

33 t

t 61

18

2x

4x

21 y 4y 74

54

u

4u

u

49

k

2k

k 35

S J Cooper

Geometry (13) Angles at a point

Work out the lettered angles for each of the following diagrams.


1. 2. 3.

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

77 a

126

103 b

195

97 c

151

105

f

3f

87

58 4g

g

117 87

5h

75

h

i

96 2i

i

p

27 69 2p

96 r

3r

3r 2r

120

d

90

40

d 72

e 67

89

2e

k

53 85

2k 45

2m

102

88

m

2m n

71

73

3n

2n

3q

65

q q

S J Cooper

Geometry (14) Angles associated with parallel lines

Work out the missing angles in each of the following triangles.


1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

k

78

j

i

129

p

m

123 t

u

v

55

a b

37

x

64

y

73 b c

a

158

e

d

f

h

g

18

n

z

x

y

104

r

q

p

71

q

s

r

46

S J Cooper

Geometry (15) Angles in a polygon

Work out the missing angles in each of the following triangles.


1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. Two sides of a triangle measured 83 and 31, what is the size of the third side?

14. In a right-angled triangle one angle is 13. What is the size of the other angle?

15. The three angles in a triangle are given by x , x + 42 and x + 51. What is the value of x?

a

27

74 c

58

56

b

40 59

i 18

42

k

k+12

36

m

m+70

2i

a

b 72

c

d 46

e f

54

g h

11

i j

124

k

l

33

S J Cooper

16. Calculate the missing angles for each of the following parallelograms

(a) (b) (c)

17. A quadrilateral has three angles of size 45, 132 and 77. What is the size of the fourth

angle?

18. Find the missing angles in the kite drawn opposite.

19. Find the interior angles for each of the following regular polygons

(a) A pentagon

(b) A nonagon

(c) A dodecagon

(d) An octagon

20. Find the size of angle x opposite.

Give a reason for your answer.

21. If the two polygons partly drawn below represent two n sided regular polygons joined at one

side how many sides does each polygon have?

68 a

c b 29

d

f

e

127 h

g

i

m

n

21 63

x

48

S J Cooper

Geometry (16) Pythagoras Theorem

Answer all the following questions, showing your working.

1. Find x 2. Find the length of PR

3. Find EF correct to 1 decimal place. 4. Find p correct to 2 decimal places

5. Find a correct to the nearest whole number

6. Find the length of the missing side, giving your answer to a suitable degree of accuracy.

8cm

6cm

x 5cm

12cm

P

Q R

22cm

18cm

p

7cm

10cm

F

E

D

5.4m

3.6m

a

9m 13m

C A

B

S J Cooper

7. Calculate the length of the diagonal in the rectangle

drawn opposite, giving your answer correct to three

significant figures.

8. Triangle DEF is isosceles

Calculate the lengths of

(i) FM (ii) DF.

9. The diagram drawn opposite represents a ladder placed

against a wall. Calculate the length of the ladder correct to

the nearest centimetre.

10. The dotted line on this map represents the journey of a ship travelling from A to D stopping at

two ports on route at B and C. Calculate the total length of this ships journey. {answers to one

decimal place}.

10cm

16cm

15cm

24cm

D

M F E

2.7m

5.4m

A

B

C

D

1 2 3 4 km

S J Cooper

Geometry (17) Pythagoras Theorem II

Answer all the following questions, showing your working.

2. Find x 2. Find the length of XY

3. Find EF correct to 2 decimal places 4. Find x correct to the nearest whole number.

5. Find d correct to one decimal place. 6. Find BC.

7. The diagram represents the front end of a garden shed.

Find the width of the shed correct to one decimal place.

16cm

20cm x 26cm

24cm

Y

X Z

27m

40m

F

E

D

29cm

23cm

x

4.5cm

11cm d

15cm A

B

C

8cm

2.2m

3.1m

2.9m

S J Cooper

8. Calculate the length of a rectangle which has width 8cm and diagonal of length 21cm. Giving

your answer to a suitable degree of accuracy.

9. Two planes are flying over the village of Colne, one directly above the other when they are

picked up by a radar station some 10km away from Colne. The distances of the planes from the

radar are given as 13km and 15 km as the diagram shows. Find the distance between the two

planes.

10. Calculate the values of x and y in the diagram below, giving your answers correct to 2 dp.

Colne

13km

15km

10km

2.4 m

8 m

6.5 m

4.8 m y

x

S J Cooper

Geometry(18) Transformations : Rotations

1. Rotate each of the following through the given angle size and direction stated.

(a) Rotate 90 clockwise centre (1, 2) (b) Rotate 180 centre (–1,2)

(c) Rotate 90 anticlockwise centre (0, –3) (d) Rotate 90 clockwise centre (4, 0)

(e) Rotate 180 Centre (3, 2) (f) Rotate 90 clockwise centre (2, –2)

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

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(g) Rotate 180 centre (–1,3) (h) Rotate 90 anticlockwise centre (2, 0)

(i) Rotate 270 clockwise centre (0, 0) (j) Rotate 270 clockwise centre (5, –1)

2. Describe the transformation which has taken place in each of the following mappings of triangle

A onto the shaded triangle.

(a) (b)

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

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3. Describe the transformation which will map triangle ABC onto triangle PQR.

4. (a) Plot the points A(1, 3), B(4, 2) and C(4, 5) and join up the points to form a triangle.

(b) Rotate triangle ABC through 90 anticlockwise, centre (1, 1) and label the image A‟B‟C‟.

5. (a) Plot the points L(–1, 3), M(–1, 0) and N(2, 2) and join up the points to form a triangle.

(b) Rotate triangle LMN through 180, centre (0, 1) and label the image L‟M‟N‟.

6. (a) Plot the points D(3, –2), E(1, –2) and F(4, 1) and join up the points to form a triangle.

(b) Rotate triangle DEF through 90 clockwise, centre (1, –1) and label the image D‟E‟F‟.

7. (a) Plot the points H(1, –3), I(1, 0) and J(4, –5) and join up the points to form a triangle.

(b) Rotate triangle HIJ through 90 anticlockwise, centre (2, –1) and label the image H‟I‟J‟.

8. (a) Plot the points S(–3, 3), T(–1, 3) and U(–2, 6) and join up the points to form a triangle.

(b) Rotate triangle STU through 180, centre (0, 1) and label the image S‟T‟U‟.

9. (a) Plot the points A(–1, –3), B(–4, –2) and C(–4, 1) and join up the points to form a triangle.

(b) Rotate triangle ABC through 90 clockwise, centre (–2, 0) and label the image A‟B‟C‟.

10. (a) Plot the points P(6, 1), Q(6, 5) and R(1, 3) and join up the points to form a triangle.

(b) Rotate triangle PQR through 180, centre (0, 3) and label the image P‟Q‟R‟.

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

A

C

B

R Q

P

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Geometry(19) Transformations : Reflections

11. Reflect each of the following in the given line.

(a) Reflect in the line 1x (b) Reflect in the line 1y

(c) Reflect in the line 1y (d) Reflect in the line 2x

(e) Reflect in the line 2x (f) Reflect in the line 0y

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

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(g) Reflect in the line 2y (h) Reflect in the line xy

(i) Reflect in the line xy (j) Reflect in the line xy

12. Describe the transformation which has taken place in each of the following mappings of triangle

A onto the shaded triangle.

(a) (b)

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

S J Cooper

13. Describe the transformation which will map triangle ABC onto triangle PQR.

14. (a) Plot the points A(1, 3), B(4, 2) and C(4, 5) and join up the points to form a triangle.

(b) Reflect triangle ABC in the line 2y and label the image A‟B‟C‟.

15. (a) Plot the points L(–1, 3), M(–1, 0) and N(2, 2) and join up the points to form a triangle.

(b) Reflect triangle LMN in the line xy and label the image L‟M‟N‟.

16. (a) Plot the points D(3, –2), E(1, –2) and F(4, 1) and join up the points to form a triangle.

(b) Reflect triangle DEF in the line 1x and label the image D‟E‟F‟.

17. (a) Plot the points H(1, –3), I(1, 0) and J(4, –5) and join up the points to form a triangle.

(b) Reflect triangle HIJ in the line xy and label the image H‟I‟J‟.

18. (a) Plot the points S(–3, 3), T(–1, 3) and U(–2, 6) and join up the points to form a triangle.

(b) Reflect triangle STU in the line xy and label the image S‟T‟U‟.

19. (a) Plot the points A(–1, –3), B(–4, –2) and C(–4, 1) and join up the points to form a triangle.

(b) Reflect triangle ABC in the line 0y and label the image A‟B‟C‟.

20. (a) Plot the points P(6, 1), Q(6, 5) and R(1, 3) and join up the points to form a triangle.

(b) Reflect triangle PQR in the line 4x and label the image P‟Q‟R‟.

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

A

C B R Q

P

S J Cooper

Geometry(20) Transformations : Enlargements

1. Enlarge the shape below by a scale factor of three centre of enlargement O. Label the image P‟.

2. Draw shape ABCD after an enlargement with scale factor 2 centre D. Label the image A1B1C1D1.

3. Enlarge the triangle LMN by a scale factor 4 centre P.

O

P

A

B C

D

P

L N

M

S J Cooper

4. The object L has been enlarged onto Image L‟.

(a) Identify the centre of enlargement and label it C.

(b) State the scale factor of the enlargement.

5. Obtain the centre and scale factor of the enlargement of the shaded shape drawn below.

L

L‟

S J Cooper

Geometry(21) Transformations : Enlargements

TAKE CARE THAT PLENTY OF ROOM IS LEFT FOR THE FOLLOWING ENLARGEMENTS!

1. For each of the following state (i) the centre of enlargement

(ii) the scale factor of the enlargement.

(a)

(b)

1 0 2 4 3 6 5 7 9 8 10

11

1

2

3

4

5

6

7

8

9

x

y

1 0 2 4 3 5 7 6 8 9 17 18 19 16 15 14 13 11 10 12 x

y

1

2

3

4

5

6

7

8

9

10

S J Cooper

(c)

(d)

2. Enlarge LMN by a scale factor of 2 centre (–1, 1); Label the image L1M1N1

-9 -8 -6 -7 -5 -3 -4 -2 -1 7 8 9 6 5 4 3 1 2 x

y

-6

-5

-4

-3

-2

-1

1

2

3

4

-1 -2 1 2 3 4 5 6 7 -3 8 x

y

-2

-1

1

2

3

4

5

0

M

L

N

1 0 2 4 3 6 5 7 9 8 10

11 x

-11 -10 -8 -9 -6 -7 -5 -3 -4 -2 -1

1

2

3

4

5

6

7

8

9

y

S J Cooper

3. Enlarge ABC by a scale factor of 4 centre (1, 2). Label the image A1B1C1

4. Enlarge the object below with centre (–3, 2) by a scale factor 3.

5. Enlarge the object by a scale factor of 3 centre of enlargement (4,5)

6. (a) Plot the points A(1, 2) , B( 3, 2) and C(3, 0) and join up the points to form a triangle ABC.

(b) Enlarge the triangle ABC by a scale factor of 3 centre (1, 3)

-1 -2 1 2 3 4 5 6 7 -3 8 x

y

-2

-1

1

2

3

4

5

0

-1 -2 1 2 3 4 5 6 7 -3 8 x

y

-2

-1

1

2

3

4

5

0

-1 -2 1 2 3 4 5 6 7 -3 8 x

y

-2

-1

1

2

3

4

5

0

A

C

B

S J Cooper

Geometry(22) Transformations : Translations

1. The diagram drawn opposite shows four triangles

drawn in different positions.

Using the vector notation describe the translation

which will map

(i) ABC onto EDG

(ii) ABC onto HIJ

(iii) ABC onto PQR

(iv) PQR onto EDG

(v) HIJ onto PQR

2. Using the drawn triangle opposite

i) draw the image A‟B‟C‟ after a translation of

ABC by

2

4

ii) draw the image A‟‟B‟‟C‟‟ after a translation of

ABC by

2

6

iii) draw the image A‟‟‟B‟‟‟C‟‟‟ after a translation

of ABC by

3

2

(iv) Describe the translation which maps A‟‟B‟‟C‟‟ onto A‟‟‟B‟‟‟C‟‟‟

3. (a) On a set of axes draw the shape STUV with coordinates S(2, 0) , T(5, 0) , U(5, 3) and

V(3, 3).

(b) Draw the image of STUV after a translation of

4

2. Label the image S‟T‟U‟V‟.

4. (a) On a set of axes draw the shape LMN with coordinates L(3, 3) , M(5, 3) , and N(4, 0).

(b) Draw the image of LMN after a translation of

4

4. Label the image L‟M‟N‟.

(c) Draw the image of L‟M‟N‟ after a translation of

2

6. Label the image L‟‟M‟‟N‟‟

-6 -5 -4 -3 -2 -1 1 2 3 4 5 x

y

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

A

B C

E

D G

H

I J

P

Q R

-6 -5 -4 -3 -2 -1 1 2 3 4 5 x

y

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

A

B

C

S J Cooper

Geometry (23) Bearings I

1. Write down the bearings of A from B for each of the following diagrams.

(a) (b)

(c) (d)

(e) (f)

(g) (h)

A

B

N

A

B

N

A

B

N

A

B

N

A

B

N

A

B

N

A

B

N

A

B

N

S J Cooper

2. Write down the bearings each of the following demonstrates

(b) (b)

(c) (d)

(f) (f)

(g) (h)

X

Y

N

K

J

G

F

N

C

D

N

T

S

N

U

V

N

N

M

N P

Q

N

S J Cooper

3. For each of the following draw a scale diagram to represent the journey, taking 1cm to

represent 2km.

(a) Starting at A Joanne travels for 10 km on a bearing of 070

(b) Starting at B James travels for 12km on a bearing of 135

(c) From C Idnan moves to point D a distance of 8km on a bearing of 056 from C, he then

changes his bearing to 145 and moves for a further 7km.

4. Using a scale of 1cm for 10km draw a scale diagram to illustrate each of the following

(a) Asif leaves home in his car on a bearing of 120 and travels for 75km, he then turns to a

new bearing of 085 and travels for a further 45km. How far is Asif from his home?

(b) A plane starting at point P moves on a bearing of 230 for 65km before changing the bearing

to 035 for 100km, reaching point Q. What is the distance PQ?

(c) A yacht leaves a harbour on a bearing of 085 and travels for 60km before changing its

bearing to 195 and travels for a further 75km. What is the shortest distance that the yacht

could have travelled?

5. Using a scale of 1cm = 10km, Simon leaves his home in Ampton and moves 46km on a bearing

of 134 until he reaches the town Burlin. At Burlin he travels for 17km on a bearing of 028 and

reaches the town Conston. How far is conston from Ampton?

6. For a lighthouse the keeper can see to ships at sea. One is on a bearing of 081 from the

lighthouse and 12km away, while the other is on a bearing of 310 from the lighthouse and

17km away.

(a) Using a scale of 1cm = 2km, draw a scale diagram to represent the information above.

(b) How far apart are the ships?

S J Cooper

Geometry (24) Bearings II

1. Town B is 16 km from town A on a bearing of 076. Town C is 25 km from Town A on a bearing

of 154. Using the scale 1 cm represents 5 km, draw a scale drawing to show Towns A, B and

C.

How far is town B from town C?

On what bearing is town B from town C?

2. A ship, S, sails a distance of 74km on a bearing of 056 and then a further 45km on a bearing of

097. Using the scale of 1 cm represents 10 km, draw a scale drawing of this journey.

How far is the ship away from its original position?

On what bearing could the ship have originally taken?

3. The insert given shows the towns of Appleton, Barton, Cotley, Dove and Eccles.

Using the diagram work out the bearing of

(a) Eccles from Appleton

(b) Cotley from Dove,

(c) Dove from Barton,

(d) Appleton from Cotley,

(e) Barton from Eccles.

4. Using the second insert a ship is spotted from the two lighthouses shown. The first lighthouse,

P, states that the ship is on a bearing of 061 while the second lighthouse, Q, states that the

ship is on a bearing of 307. Using a suitable construction identify on the insert the position of

the ship.

S J Cooper

Geometry (24) Inserts

Insert 1

Insert 2

Appleton

Q

Barton

Dove

Eccles Cotley

P

S J Cooper

Geometry (25) Introduction to Trigonometry

Exercise With each of the right-angled triangles below, write the name of each lettered side.

3. 2. 3.

4. 5. 6.

7. 8. 9.

10.

Hypotenuse

Adjacent

Opposite

Remember

a b

c

f

e d

h g

i

l j

k

x

w

y

d

e

c

u

t

v

a

b

z

n p

m

q s

r

S J Cooper

Geometry (26) The Sine ratio

1. For each of the following triangles below:

(i) write down the length of the opposite side

(ii) write down the length of the hypotenuse

(iii) calculate the ratio Hypotenuse

Opposite

Give your answer correct to 3 decimal places where necessary.

Example

(i) Opposite = 2cm

(ii) Hypotenuse = 4cm

(iii) 5.04

2

Hypotenuse

Opposite

a. b. c.

d. e. f.

Look on your calculator for a button , we use this if we want to find the Sine of the angle

(“Sin” is short for sine)

Example

To find the sine of 50, Press and then the buttons followed by

2cm

4cm

30

2cm

6cm

19.47

12cm 2.5cm

12.02

3cm 4cm

48.6

5cm

1.5cm

17.46

4.9cm

2.2cm

24.04

5.4cm 5cm

4cm

53.13

Sin

Sin

5 0 =

S J Cooper

2. (a) Find the sine of each of the angles in the triangles of question 1, giving your answer to 3

decimal place.

(b) Compare each of your results to your answers to part (iii) in question 1.

What do you notice?

(c) Complete the equation opposite

Exercise

For each of the following triangles, find the length of the lettered side, giving your answers correct

to 1 decimal place.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10.

x

Sin x = ……

6cm 54

a

2cm

65

b

7cm

32

c

6mm

18

d

5cm

79

e

4m

71

f

9m

30

g

6cm

19

h

3cm

46

i

4cm

52

j

S J Cooper

Geometry (27) The Cosine & Tangent ratio

1. For each of the following triangles below:

a. write down the length of the adjacent side

b. write down the length of the hypotenuse

c. calculate the ratio Hypotenuse

Adjacent

Give your answer correct to 3 decimal places where necessary.

a. b. c.

d. e. f.

Look on your calculator for a button , we use this if we want to find the Cosine of the angle

(“Cos” is short for cosine)

2. (a) Find the sine of each of the angles in the triangles of question 1, giving your answer to 3

decimal place.

(b) Compare each of your results to your answers to part (iii) in question 1.

What do you notice?

(c) Complete the equation opposite

5cm

4cm

36.87

6cm

2.5cm

65.38

4.9cm

68.68

5.4cm

8cm

4.5cm 55.77

Cos

x

Cos x = ……

3cm

4.5cm

48.19

4.5cm

4cm

27.27

S J Cooper

Exercise


to 2 decimal place.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10.

Look on your calculator for a button , we use this if we want to find the Tangent of the

angle (“Tan” is short for Tangent). Using the two triangles drawn below find which ratio is equal to

the tangent of the angle.

6cm

24

a

8cm

45

b 7cm 18

c

6mm

18

d

5m

32

e 7.2cm 38

f

8cm

24

h

12cm

29

i

2.4m

52

j

6cm

66

g

Tan

5cm

36.87

4cm

3cm 6.5cm

22.62

6 cm

2.5cm

S J Cooper

Exercise


to 2 decimal place.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10.

6cm

42

a

7m 71

c

3m

26

d

8.3cm

67

e 6m

18

f

3.2cm

8 h

8cm

74

i

65mm

24

g

7cm

31

b

5cm 58

j

S J Cooper

Geometry (28) Using The three trig ratios

1. For each of the following questions find the length of the missing sides. Giving your answers

correct to one decimal place.

(a) (b) (c)

(d) (e) (f)

2. For the triangle drawn opposite find the length of

(i) AB (ii) BC

3. For the triangle drawn opposite find the lengths of

(i) PR (ii) QR

4. In triangle ABC, Angle C = 90, Angle A = 74 and AC = 19cm. Find the length of BC.

5. In triangle PQR, Angle Q = 90, Angle P = 37 and PR = 3.2cm. Find the length of PQ.

4cm

32

x

8cm

54

x

9cm

58

x

4.5cm

41

x

2.4m

49

j

5cm

70

x

9cm

62

C

B A

8cm

52

P

R

Q

S J Cooper

6. In triangle LMN, Angle M = 90, Angle L = 81 and LN = 14cm. Find the length of MN.

7. In triangle DEF, Angle F = 90, Angle D = 21 and DF = 3.2cm. Find the length of EF.

8. In an isosceles triangle ABC, angle A is 47 and the length AB = BC = 10cm. Calculate the

length of AC.

9. For the diagram drawn below find the lengths of x and y

10. Find the height of these stairs correct to one decimal place.

11. A kite is flying at a height which makes an angle of 30 to the horizontal. If the length of string is

42 metres in length, how high is the kite?

12. The diagram below represents the cross section for the framework of a tent.

Calculate correct to one decimal place the heights of the points A, B and C from the ground.

15cm

32

x 20 y

35

3.6m

h

75

68 60

8.5m

11.4m

10.2m

3.2m

A

B

C

S J Cooper

Geometry (29) Finding an angle using trigonometry

For each of the following find the size of the missing angle.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13cm

x

5cm 4.2cm

2.4cm

x

10cm

25cm x

21cm

14cm

x

8cm 7cm

x

8cm

7cm

x

6cm

x

3.7cm

27cm

x

18cm

3cm

4.2cm

x

10.3cm

7.4cm

x 14m

9m

x

6.5m

3.4m

x

S J Cooper

13. In triangle ABC, angle A = 90, AC = 60 cm and BC = 72cm. Find angle C.

14. In triangle PQR, angle Q = = 90, PQ = 12cm, QR = 14cm. Find angle P.

15. In triangle XYZ, angle Z = 90, XY = 16m, XZ = 8m. Find angle Y.

16. In triangle LMN, angle M = 90, LM = 1.6cm and MN = 0.9cm. Find angle N.

17. The two equal sides of an isosceles triangle are 15cm long. If the height of the triangle is 7cm,

find the size of the angles in the triangle.

18. An isosceles triangle has sides 20cm, 20cm and 10cm. Find the size of all angles in this

triangle.

19. The sketch drawn below represents a rope slide from a cliff to the beach below. The cliff is a

height of 50m and the rope is set at 150m from the bottom of the cliff. Find the angle that the

rope makes with the beach.

20. the diagram represents a lighthouse

of height 135mand a boy standing at

point P, 375m away. What is the

angle of elevation from the boy to

the top of the lighthouse?

150m

50m

x

375m

S J Cooper

Geometry(30) Circle theorems Work out the lettered angles in each of the following diagrams

(1) (2)

(3) (4)

(5) (6)

(7) (8)

50

a

78

b

15

c

17

d

60

e

g

46

h

37

x

95

47

40

x

98

i

y

S J Cooper


1. 2.

3. 4.

5. 6.

7. 8.

b

43

e

25

a

70 f

120

c

n

150

k

98

m

37 h

r 28

y

95 s

t

x

S J Cooper


1. 2.

3. 4.

5. 6.

7. 8.

b

43

a

146

d

78 x

48

e 40 d 29

n

38

u

54

m

p

v

a

b

c

y

z

f

q

S J Cooper


1. 2.

3. 4.

5. 6.

7. 8.

b

78

c

115

a

84 64

d

i

80

e

50

h g

f

p

25

r 70

q

n

54

82

m k

21

u 95

y

74

s

t

x

110

w

S J Cooper


1. 2.

3. 4.

5. 6.

49

a

c

36

b

18

e

28 n

19 m

s

t

u

r

30

70

S J Cooper

Geometry (35) Circle Theorems

Work out the lettered angles in each of the following diagrams

(1) (2)

(3) (4)

(5) (6)

(7) (8)

24

80 110

15

74 68

32

85

24

40

65

25

14

n

a b

c

d

e

f

g

h i

k m

S J Cooper

Geometry (36) The Sine rule

1. Work out the lettered side for each of the following:

(a) (b)

(c) (d)

(e) (f)

2. In triangle STU, 5.7ST cm, 45ˆ UST and 30ˆ STU

Work out the length of TU.

3. In triangle LMN, 7.4LM cm, ˆ 54LMN and ˆ 78MLN

Work out the length of MN.

a

7 cm

40º

30º

95º

20º

125º

10º 70º

35º

50º 45º

68º

27º

b

c

d

e

f

17 cm

3.7 cm

6 m

8.2 cm

12 cm

S J Cooper

4. Work out the lettered angle for each of the following:

(a) (b)

(c) (d)

(e) (f)

5. In Triangle LMN, 6LM cm, 35ˆ NLM and 7LN cm.

Work out the size of NML ˆ .

6. In the triangle XYZ when 5.3XZ cm, 8.5YZ cm and 68ˆ ZYX

Work out the size of ˆXZY .

7. In the triangle STU when 3.8ST cm, 9.4TU cm and 29ˆ STU

Work out the size of ˆTSU .

A

7 cm

70º

20º

88º

115º

69º

35º

B

C

D

E

F

24 cm

5.4 cm

3 m

7.1 cm

30 cm

6 cm 13 cm

6.3 cm

9 m

7.6 cm

23 cm

S J Cooper

8. For each of the triangles below find the lengths or angles required.

(a) (b)


(c) (d)

Find Angle N Find Angle M

70º

9.7cm 10cm

42

6.5cm 8cm

E 36

4cm

122

F

N

94

5.4cm

8.1cm

M

S J Cooper

Geometry (37) The Cosine rule


(a) (b)

(c) (d)

(e) (f)

10. In the triangle LMN, 8LM cm, 35ˆ NLM and 5LN cm.


11. IN the triangle XYZ, 5.3XY cm, 8.5YZ cm and 68ˆ ZYX

Work out the length of XZ.

a

7 cm 20º

75º

115º

77º

124º

50º

b

c

d

e

f

9.4 cm

5.7 cm 5 m

16 cm

29 cm

6 cm 8.5 cm

7.3 cm

13 m

8 cm

28 cm

S J Cooper


(a) (b)

(c) (d)

13. In triangle ABC sides are AB = 8cm, AC = 6cm and BC = 4cm.

Work out the size of CBA ˆ .

14. In triangle HIG sides are HI = 6.4cm, HG = 7.1cm and IG = 4.5cm.

Work out the size of GIHˆ .

15. In triangle ABC where AB = 5cm, BC = 6cm and AC = 4.3cm. Work out the size of the largest

angle.

A

7 cm

6 cm

8 cm 14 cm

10 cm

15 m

5 m

9 m

8 m

B

C

D

14 cm

11 cm

6 m

S J Cooper

Geometry(38) Area continued

1. Calculate the area for each of the following triangles

(a) (b) (c)

(d) (e)

2. (a) Work out the size of angle A in the triangle below

(b) Hence find the area of triangle ABC

3. (a) Work out the size of angle D in the triangle below.

(b) Hence find the area of triangle DEF

6cm

9cm

60°

7.5cm 5.7cm

37°

8.3cm

1.8cm

115°

34cm

57cm

40°

19cm

21cm 17°

A

B

C 20°

10cm

15cm

7cm

11cm 8cm

D

E

F

S J Cooper

4. Work out the area of the triangles drawn below

a) (b)

5. The diagram drawn is of a cube with a corner cut out. Given that all measurements are in

centimetres find the surface area of the cube.

6. Find the area of the shape drawn below. [Be careful..this involves a lot of previous knowledge]

20cm

7cm

20cm

17cm

25°

80°

10cm 10cm

10cm 10cm

10cm

3 4

5

9

9 9

S J Cooper

Geometry (39) Volume of a Pyramid/Sphere

8. For each of the following work out the volume, where appropriate giving your answers to 2

decimal places.

(i) (ii) (iii)

(iv) (v) (vi)

9. The diagram shows the cork top of a bottle

with dimensions given. Find its volume.

10. The diagram is of a garden pot with square base 50cm and top

60cm. Find its volume.

5cm

5cm

7cm

9cm

5cm

10cm

Area = 105 cm2

8 cm

18 m

15mm

3.5mm

1cm

1.4cm

2.2cm

5cm

95cm

55cm

S J Cooper

11. Find the volume of the shape drawn below

12. The volume of a square based pyramid with height 12cm is 144 cm3. Find the length of the side

of the square.

13. A cone has volume 108 m3. Find its radius when its height is 4m.

20 cm

70 cm

S J Cooper

Geometry(40) Surface Area

1. Calculate the surface area for each of the following shapes

(a) (b) (c)

(d) (e) (f)

2. Work out the total surface area of the hemisphere drawn below.

3. Calculate the surface area of the shape drawn below.

(a) (b)

12cm

4cm 7cm

5cm

20cm

15cm

8cm

24cm

9cm

40cm

40cm

25cm

10 cm

18mm 15mm

5mm

4mm

3mm

3mm

10mm

S J Cooper

Geometry (41) Similar Shapes

1. In each of the following finds the length of the lettered side, given that each pair of shapes

are similar.

(a)

(b)

(c)

2. Show that triangle ABC is similar to triangle ADE

Hence work out the length of (i) DE

(ii) CE

3. Given that the two rectangles drawn are

similar find the height of the rectangle

labelled A.

Hence find the areas of the rectangles A and B.

Deduce the relationship between the areas of A and B and the length ratios of A and B

5 cm

9 cm

a

27 cm

b

6 cm

9cm

27

c

7 cm

8 cm 6

d

4cm

A

E C

D

B

10cm

9cm

3cm

7cm

2 8

16

S J Cooper

Geometry (42) Similar Shapes II

1. In each of the following finds the area of the shape, given that each pair of shapes

are similar.

a)

b)

c)

d)

2. A triangle has sides 5cm, 12cm and 13cm, and has an area of 30cm2. A similar

triangle has an area of 120cm2. Find the lengths of each side of the larger triangle. 3. Two similar cones have heights 4cm and 8cm respectively. If the volume of the

larger cone is 56cm3, find the volume of the smaller cone.

5 cm2

9 cm

A

27 cm

8 cm

20 cm

32 cm2 B

15 cm 6 cm 45 cm

2 C

15 cm

D

5 cm

45 cm

S J Cooper

4. Two similar spheres have masses of 24kg and 648kg respectively. If the radius of

the smaller sphere is 5cm find the radius of the larger sphere.

5. In the diagram below the two cylinders are similar. Find the length of the lettered

side.

6. In triangle XYZ a line AB parallel to YZ is drawn such that AX = 2cm. Given that AY

= 3cm and the area of triangle XYZ is 50cm2, find the area of the trapezium ABZY.

7. Find the volume of the larger solid of the two drawn below, given that both solids

are similar.

8. Two similar solids have surface areas 20m2 and 45m2 respectively, given that the

mass of the smaller solid is 56kg find the mass of the larger solid.

9. Two similar spheres have masses of 128 kg and 250kg, respectively. Given that the

surface are of the larger sphere is 75cm2, find the surface are of the smaller

sphere.

192cm3 3cm3 3cm x cm

A

Y Z

B

X

8cm

12cm

24cm3

S J Cooper

Geometry(43) Vectors

1. Write the components of each vector in the diagram below.

2. Write down in component form each of the following vectors

3. By drawing a suitable diagram or otherwise state the vector which joins the

points A(1, 2) and B(4, 6) together.

4. Which vector moves the point C(-1, 4) to the point D(5, -3)?

5. Draw suitable diagrams to illustrate each of the following vectors. Label each

vector accordingly.

a)

4

1a b)

2

3b c)

2

5AB d)

5

4LM

6. Given

2

6a and

3

1b work out the vector ba . Represent your answer on a

suitable diagram.

7. Find the values of the missing letters in each of the following additions.

a)

6

35

1 b

a b)

5

3

7

2 e

d c)

3

8

4

1 m

n

a b c d

A B

C

D

E

F

G

H

S J Cooper

8. Use the diagram given to find the appropriate component form for the vector

equivalent to

a. yx

b. zyx

c. azyx

9. Given

0

4a and

3

2b work out the vectors

a) ba 2

b) ba

c) ba 32

d) ba 22

1

x

y

z

a

S J Cooper

A

E

B

D

F

C

a

b

L M

N

Q P

Geometry(44) Vector Geometry

1. Given the vectors a and b below draw diagrams to represent each of the

following vectors

a) ba b) ba c) ba 2 d) ba 2 e) ab 32

2. In the parallelogram ABCD drawn opposite E

and F are the midpoints of AB and CD

respectively.

If aAD and bAE , write in terms of

a and b

(i) AB (ii) AF (iii) AC (iv) BD

3. In the triangle LMN points P and Q are the

midpoints of the lines LN and MN respectively.

Given that aLN and bLM m write in terms

of a and b

(i) LP (ii) MN (iii) NQ (iv) LQ

a b

S J Cooper

X Y

Z

T

4. In the triangle XYZ the point T is such that YT=3ZT.

Given that pXZ and qXY , express in terms of p

and q

(i) YZ (ii) YT

(iii) XT

5. The diagram below consists of three equilateral triangles joined together.

Work out each of the following vectors

a) AD (b) AB (c) OB (d) AC

6. OABCDE is a regular hexagon with OA represented by the vector a and OE

represented by the vector e. Find the vectors representing

(i) AB (ii) OC (iii) AD

O

A

a

B

C D d

S J Cooper

O

P

B C

A

Q

O

N

M

L

R

Q P

S

O

P

A

C

Q

c

a

B

Geometry(45) Vectors concluded

1. Relative to O the position vectors of A and B are a and b. Point P is a point on AB

such that AP = 2PB

Find in terms of a and b

(i) AB (ii) AP (iii) OP

2. OACB is a square with aOA and bOB

P is a point on AC such that AP : PC = 1 : 3 and Q

is on OB such that OQ : QB = 3 : 1.


(i) OQ (ii) OP

3. OLMN represents a kite with aOL , bON and cLM

Points P, Q, R and S are the midpoints of the lines

LM, MN, ON and OL respectively.

a) Find in terms of a, b and c

(i) NM (ii) SR (iii) PQ

b) Comment on your finding in part (a)

4. OABC is a parallelogram with aOA and

cOB

P is a point on AC such that 3

1

PC

AP and

Q is the midpoint of BC.


(i) OP (ii) OQ

O

A

B

P

S J Cooper

S J Cooper

Geometry (46) Special Curves

1. (a) Copy and complete the table below for the graph of xy sin

x 0 30 60 90 120 150 180 210 240 270 300 330 360

y 0.5 0.87 -0.86

(b) On Graph paper draw the graph of xy sin

(c) Use your graph to solve each of the following equations

(i) 75siny (ii) 8.0sin x (iii) 2.0sin x

2. (a) Copy and complete the table below for the graph of xy cos

x 0 30 60 90 120 150 180 210 240 270 300 330 360

y 0.5 -0.86 0

(b) On Graph paper draw the graph of xy cos

(c) Use your graph to solve each of the following equations

(i) 6.0cos x (ii) 6.0cos x

3. On the calculator there is a button ex, meaning exponential of x.

a) Use this button to complete the table below.

b) On graph paper draw the graph of xey

4. Given that 64.040sin state another angle which would give the answer 0.64.

5. Given that 17.0100cos state another angle which would have given the answer -

0.17.

x -3 -2 -1 0 1 2 3

y 0.14 2.72

S J Cooper

Geometry (47) 3D Coordinates 1. For each of the following write down the coordinates of the vertices

(a) (b)

(c) (d)

(e) (f)

2. Each of the blocks in the diagram below has an edge of one unit. Write down the coordinates of A, B, C, D, E,

F, G and H

S J Cooper

3. A, B and C represent the vertices of a cuboid and are the points (4, 0, 0) , (0, 2, 0) and (0, 0, 3) respectively.

Work out the coordinates of the other vertices for this cuboid.

4. Given that each cube has an edge of one unit, work out the length of AD in the diagram below

S J Cooper

Geometry (48) 3D problems

1. The diagram below represents a cuboid with dimensions AB = 12 cm, BC = 6cm and AH = 5cm.

(a) Work out the lengths AG and AF

(b) Work out the angle AG makes with the line AB

(c) Work out the angle AF makes with the plane

ABCD.

2. The diagram shows a triangular prism with

AB = 8cm, AF = 6cm and BC = 15cm.

(i) Work out the lengths of FB and FC

(ii) What is the length of the diagonal in

the rectangle ABCD?

(iii) Find the angle FBA ˆ

(iv) Work out the angle made between

FC and the plane ABCD.

3. The diagram opposite is of a square based pyramid with side 7cm

and

slanted edge 9cm.

Work out

a) The length of the diagonal AC

b) The height of the Pyramid, EF

c) The angle EAB ˆ

d) The angle EB makes with the base ABCD

4. The diagram is of a “wedge” used for keeping a door open.

The base is square with AB = 12cm.

Angle 20ˆ FBA

Work out

a) The height AF

b) The length of the diagonal BE

c) The angle DBE ˆ

S J Cooper

5. ABCDEFGH is a cuboid with dimensions 5cm, 6cm, 14cm as shown.

Calculate the size of angle BEG.

6. ABCDEF represents the roof of a building.

The base ABCD forms a rectangle with dimensions 12m by

4m.

ABF and DCE are identical isosceles triangles with slanted

edge 6m. G and H are the midpoints of AB and DC

respectively.

Work out

(a) The lengths FG and EA

(b) The perpendicular length from point F to the base ABCD

(c) The angle HGF ˆ

(d) Angle ADE ˆ

7. The picture is of one of the largest pyramids in Egypt, the pyramid of Giza. As one of the oldest seven

wonders of the world its height was approximately 146 m tall and the square base is approximately 240 m

long.

Work out

a) The length of the diagonal on the base.

b) The length of the slanted edge from base to

the top.

c) The angle made between the slanted edge

and the diagonal.

S J Cooper

Geometry (49) The General Triangle


(a) (b)

(c) (d)

2. In triangle STU, 5.7ST cm, 45ˆ UST and 30ˆ STU

Work out the length of TU.

3. In triangle LMN, 7.4LM cm, ˆ 54LMN and ˆ 78MLN



(a) (b)

a

7 cm

40º

30º

95º

20º

125º

10º 70º

35º

b

c

d

17 cm

3.7 cm

6 m

A

7 cm

70º

20º

B

24 cm 6 cm

13 cm

S J Cooper

(c) (d)

5. In Triangle LMN, 6LM cm, 35ˆ NLM and 7LN cm.

Work out the size of NML ˆ .

6. In the triangle XYZ when 5.3XZ cm, 8.5YZ cm and 68ˆ ZYX

Work out the size of ˆXZY .

7. In the triangle STU when 3.8ST cm, 9.4TU cm and 29ˆ STU

Work out the size of ˆTSU .

8. For each of the triangles below find the lengths or angles required.

(a) (b)



(a) (b)

88º

115º

C

D

5.4 cm

3 m

6.3 cm

9 m

42

6.5cm 8cm

E 36

4cm

122

F

a

7 cm 20º

75º

b

9.4 cm 6 cm

8.5 cm

S J Cooper

(c) (d)

10. In the triangle LMN, 8LM cm, 35ˆ NLM and 5LN cm.


11. IN the triangle XYZ, 5.3XY cm, 8.5YZ cm and 68ˆ ZYX

Work out the length of XZ.


(a) (b)

(c) (d)

115º

77º

c

d

5.7 cm 5 m

7.3 cm

13 m

A

7 cm

6 cm

8 cm 14 cm

10 cm

15 m

5 m

9 m

8 m

B

C

D

14 cm

11 cm

6 m

S J Cooper

13. In triangle ABC sides are AB = 8cm, AC = 6cm and BC = 4cm.

Work out the size of CBA ˆ .

14. In triangle HIG sides are HI = 6.4cm, HG = 7.1cm and IG = 4.5cm.

Work out the size of GIHˆ .

15. In triangle ABC where AB = 5cm, BC = 6cm and AC = 4.3cm. Work out the size of the

largest angle.

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