Thomas Whitham Sixth Form

YYEEAARR 1111

Higher Tier

GGEEOOMMEETTRRYY

TTrriiggoonnoommeettrryy && PPyytthhaaggoorraass

TThheeoorreemm

Area and Volume

Similar Shapes

Constructions

Angles

1

Geometry (1) Pythagoras Theorem

Answer all the following questions, showing your working.

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

3. Find d correct to one decimal place. 4. Find BC.

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

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

6. The diagram drawn opposite represents a ladder placed

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

nearest centimetre.

22cm

18cm

p

7cm

10cm

F

E

D

4.5cm

11cm d

15cm A

B

C

8cm

2.2m

3.1m

2.9m

2.7m

5.4m

2

7. 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}.

8. 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.

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

A

B

C

D

1 2 3 4 km

Colne

13km

15km

10km

2.4 m

8 m

6.5 m

4.8 m y

x

3

10. The diagram below is of a triangular prism with triangle ABC a right angled triangle.

Furthermore AB = 4cm, BC = 7cm and BE = 10cm.

Find the length of (a) AC (b) AD, giving your answers correct to 2 decimal places.

11. The diagram below represents a cuboid. Find the length of the diagonal PV, giving your answer correct

to the nearest whole number.

A

D

E

F

C

B

P

V W

S

T

R

U

17cm

9cm

21cm

4

Geometry (2) Right angled Trigonometry

Exercise 1

For each of the following triangles, find the length of the lettered side, giving your answers correct to 2

decimal place.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10.

4cm

32

x

8cm

54

x

9cm

58

x

4.5cm

41

x

2.4m

49

j

5cm

70

x

7m 71

c

7cm

31

b

2.4m

52

j

6cm

66

g

5

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.

Exercise 2

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

1. 2. 3.

4. 5. 6.

13cm

x

5cm 4.2cm

2.4cm

x

21cm

14cm

x

8cm 7cm

x

6cm

x

3.7cm

75

68 60

8.5m

11.4m

10.2m

3.2m

A

B

C

6

7. 8. 9.

10. 11. 12.

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.

8cm

7cm

x

27cm

x

18cm

3cm

4.2cm

x

10.3cm

7.4cm

x 14m

9m

x

6.5m

3.4m

x

7

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

8

Geometry (3) 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

b 17 cm

125º

10º

c

3.7 cm

70º

35º

d

6 m

50º 45º

e

8.2 cm

68º

27º

f

12 cm

9

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

6 cm

20º

B

24 cm 13 cm

88º

C

5.4 cm

6.3 cm

115º

D

3 m

9 m

69º E

7.1 cm 7.6 cm 35º

F 30 cm

23 cm

10

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

(a) (b)

Find Angle E Find length F

(c) (d)

Find Angle N Find Angle M

42

6.5cm 8cm

E 36

4cm

122

F

70º

9.7cm 10cm

N

94

5.4cm

8.1cm

M

11

Geometry (4) The Cosine Rule

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

(a) (b)

(c) (d)

(e) (f)

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

Work out the length of MN.

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

Work out the length of XZ.

a

7 cm 20º

6 cm

75º

b

9.4 cm

8.5 cm

115º

c

5.7 cm

7.3 cm 77º d

5 m

13 m

124º

e

16 cm

8 cm

50º

f

29 cm

28 cm

12

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

(a) (b)

(c) (d)

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

Work out the size of CBA ˆ .

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

Work out the size of GIHˆ .

7. 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 B

14 cm

15 m

5 m

C

11 cm

9 m

8 m

D

6 m

13

Geometry (5) Area

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

(a) (b) (c)

(d) (e) (f)

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

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

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

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

7. Find the area of the semicircle drawn opposite, giving

your answer to 2 decimal places.

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

answer correct to 1 decimal place.

8cm

5cm

12cm 23mm

21mm

19mm

9cm

13cm

7cm

8cm

7cm

15cm

12cm

10cm

3m

13m

3.7m 7.2cm 75m 60km

36 cm

100 cm

32 cm

14

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

figures]

(a) (b) (c)

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

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

places.

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

12. Find the perimeter of the semicircle drawn opposite,

giving your answer to 2 decimal places.

13. (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?

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

nearest whole number.

15. What is the diameter of a circle whose circumference is 24cm? [answer correct to 1 decimal place].

16. What is the circumference of a circle whose area is 60cm2? { answer to the nearest whole number]

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

12cm 6cm 5m 12m

7m

6.1m 2.3cm 112m 38m

50 cm

150m

70m

15

Geometry (6) 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

16

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

17

Geometry (7) Volume of a prism

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

places.

(i) (ii) (iii)

(iv) (v)

(vi) (vii) (viii)

(ix) (x)

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 ?

32cm

15cm

8cm

17m

16m

24m

3m

16m

4cm

7cm

5cm

0.5m

11m

4m

7m

5m

12m

6.5m

7.5m

2cm

7cm

8.9cm

7.8cm

8m

12cm

15cm

18

4. 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)

5. 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.

10

5

7

6

6

9

12

6

7

5

8

7 15

10

19

17

9

8

8

10

20

15

19

Geometry(8) 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

20

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

21

Geometry(9) Arc Length and area of a sector

1. For each of the following work out the area of the sector shaded.

a) b) c)

d) e) f)

2. For each of the following work out the arc length XY.

a) b) c)

3. Work out the perimeter of the shape given below.

5cm

120°

7cm 70°

3.7cm

115°

21m

270°

130°

10mm

325°

3.4m

6m 123°

7.8cm

149°

1.4m

28°

95° 10cm

15cm

15cm

5.6cm

22

Geometry (10) Volume of a Pyramid/Sphere

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

places.

(i) (ii) (iii)

(iv) (v) (vi)

2. The diagram shows the cork top of a bottle

with dimensions given. Find its volume.

3. 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

23

4. Find the volume of the shape drawn below

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

square.

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

20 cm

70 cm

24

Geometry(11) 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

25

Geometry (12) 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

26

Geometry (13) 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.

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 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

27

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

28

Geometry (14) Constructions Name………………..

Do the following constructions within the spaces provided [practice first]

1. Perpendicular bisector of AB. 2. Bisector of angle ABC

3. Perpendicular bisector of PQ. 4. Bisector of angle LMN.

5. Bisector of angle EDF. 6. The perpendicular from P to AB.

7. The perpendicular from X to JK. 8. Perpendicular bisector of JK.

A

B

A B

C

P

Q

L M

N

D F

E

A B

P

J

K

X J

K

29

9. PQR = 60. Label Point R. 10. Angle BAC =90. Label point C

11. Angle XYZ = 30. Label point Z. 12. Angle QPR = 45. Label point R.

13. Angle MLN = 120. Label point N. 14. Angle EFG = 15. Label point G.

15. Angle JKL = 75 Label point L.

16. Angle STU = 150. Label point

P Q A B

P Q

L M E F

X Y

J K T U

30

Geometry (15) 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 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

31

Geometry (16)

Constructions of triangles without protractor

REMEMBER DO NOT REMOVE ANY CONSTRUCTION LINES OR ARCS.

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

Measure and write down the size of CAB ˆ .

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

Measure and write down the size of FED ˆ

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

Measure and write down the length of XZ.

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

Measure and write down the length of ST.

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

Measure and write down the length of MN.

11. Construct each of the following triangles

(a) (b)

(c)

8cm

6cm

7.5cm

30

135 30

5cm

32

12. (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.

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

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

State the length of the diagonal AC.

15. Construct the trapezium below.

60

9cm

6cm

33

Geometry(17) Transformations

1. (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‟.

2. (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‟.

3. (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‟.

4. (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‟.

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) 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‟.

10. (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‟‟

34

11. (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)

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

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

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

-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

35

15. 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

36

(c)

(d)

-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 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

37

Geometry( 18) 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

38

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

39

A

E

B

D

F

a

b

L M

N

Q P

X Y

Z

T

Geometry( 19) 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

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

a b

40

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

41

O

P

B C

A

Q

O

N

M

L

R

Q P

S

O

P

A

C

Q

c

a

B

Geometry(20) 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.

Find in terms of a and b

(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.

Find in terms of a and b

(i) OP (ii) OQ

O

A

B

P

42

Geometry (21) 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

43

Geometry(22) 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

17

d

47

60

e

40

15

c

95 x

g

98

46

h

i

37

x

y

44

Geometry(23) Circle theorems Work out the lettered angles in each of the following diagrams

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

45

Geometry(24) Circle theorems Work out the lettered angles in each of the following diagrams

1. 2.

3. 4.

5. 6.

7. 8.

b

43

a

a

146 b

d

78

c

x

48 y

z

e 40

f

d 29

n

38

m

p

q u

54

v

46

Geometry(25) Circle theorems Work out the lettered angles in each of the following diagrams

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

47

Geometry(26) Circle theorems Work out the lettered angles in each of the following diagrams

1. 2.

3. 4.

5. 6.

49

a

c

36

b

18

e

28 n

19 m

s

t

u

r

30

60

48

Geometry (27) 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