THOMAS WHITHAM SIXTH FORM

Transcription

1 THOMAS WHITHAM SIXTH FORM Mathematics Geometr GCSE Unit 3 t h o m a s w h i t h a m. p b w o r k s. c o m

2 Geometr (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 AB ˆ C. 2. Draw a triangle LMN where LM 6cm, Measure and write down the size of ˆ 35 M LN and LN 7 LM ˆ N. cm. 3. Draw the triangle XYZ when XY 3. 5cm, YZ 5. 8 cm and Measure and write down the length of XZ. ˆ X YZ Draw a triangle HIG whose sides are HI = 6.4cm, HG = 7.1cm and IG = 4.5cm. Measure and write down the size of Hˆ IG. 5. Draw the triangle STU when ST 8. 3cm, TU 4. 9 Measure and write down the length of ST. cm and ˆ U TS Draw accurate diagrams for each of the triangles below and find the lengths required. (a) (b) 8cm F E 10cm cm 122 Find Angle E Find length F (c) N (d) 10cm 10cm 8.1cm 7cm 94 M 5.4cm Find Angle N Find Angle M

3 Geometr (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 ABC 60. Measure and write down the size of BA ˆ C. 8. Draw a triangle DEF where DE 5cm, DF 7 cm and Measure and write down the size of DE ˆ F ˆ E DF Draw the triangle XYZ when XY 4. 2 cm, YZ 6. 3 cm and Measure and write down the length of XZ. ˆ X YZ Draw the triangle STU when ST 7. 5 cm, Measure and write down the length of ST. ˆ TSU 45 and ˆ U TS Draw the triangle LMN when LM 4. 7 cm, Measure and write down the length of MN. ˆ LMN 45 and M LN ˆ Construct each of the following triangles (a) (b) 6cm 8cm 7.5cm 30 (c) cm 13. (a) Construct triangle ABC where AB = 5cm, BC = 6cm and AC = 4.3cm. (b) Bisect the side given b 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.

4 14. Using ruler and compasses onl construct a rectangle with dimensions 7cm b 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. 6cm 9cm 60

5 Geometr (5) Area and perimeter 1. Find the area for each of the following shapes without the use of a calculator. (a) (b) (c) 26cm 23m 23cm 14cm 23m 12cm (d) (e) (f) 9cm 12cm 8cm 23mm 19mm 7cm 5cm 21mm 13cm (g) (h) 12cm (i) 7cm 10cm 13m 8cm 15cm 3m 2. A rectangle of area 216 m 2 has length 18m what is its width? 3. A triangle with base length 15cm has an area of 210cm 2, 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 b 6.4m. If a rug with dimensions 4.6m b 3.7m is placed onto 6.4m 4.6 m 3.7 m the floor calculate the area of flooring not covered b the rug. 7.5m 6. A photograph 27cm b 25cm is placed into a frame with dimensions 34cm b 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?

6 Geometr (6) Area & perimeter II 1. Calculate the area for each of the following circles, giving our answers correct to 1 decimal place. (a) (b) (c) (d) 5m 14cm 14cm 54m 2. Calculate the area of each of the following circles giving our answers correct to 2 decimal places. (a) (b) (c) (d) 3.7m 7.2cm 75m 60km 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 54cm 2 find its radius correct to 2 decimal places. 5. Find the area of the semicircle drawn opposite, giving our answer to 2 decimal places. 36 cm 6. Find the area of the shape opposite, giving our 32 cm answer correct to 1 decimal place. 100 cm 7. Calculate the shaded area for each of the following shapes. [giving our answers correct to 2 significant figures] (a) (b) (c) 6cm 12cm 5m 12m 7m 8. A circle has an area of 125 cm 2. Calculate the length of its radius, giving our answer to 2 decimal places.

7 Geometr (7) Circumference of a circle 1. Calculate the circumference of each of the following circles, giving our answers correct to one decimal place. (a) (b) (c) (d) 7cm 23m 19cm 43m 2. Calculate the circumference of each of the following circles, giving our answers correct to 2 decimal places. (a) (b) (c) (d) 6.1m 2.3cm 112m 38m 3. Find the perimeter of the semicircle drawn opposite, giving our answer to 2 decimal places. 50 cm 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 70m diameter 70m. (b) What is the length of one complete lap? 150m (c) How man laps (approximatel) must an athlete run in a race of m? 5. A biccle wheel has diameter 75cm. Calculate its circumference, giving our answer correct to the nearest whole number. 6. What is the diameter of a circle whose circumference is 24cm? [Give our answer correct to 1 decimal place]. 7. What is the circumference of a circle whose area is 60cm 2? {give our answer correct to the nearest whole number] 8. Which has the greatest perimeter, a circle with radius 6cm or a square with side 5cm?

8 Geometr (8) Area & perimeter of irregular shapes 1. Work out the area and perimeters for each of the following irregular shapes. (a) (b) (c) 4m 7cm 12cm 13cm 6cm 4cm 7cm 14cm 23cm 23cm 21m 7m 19m 7m 12m 2. Work out the area for each of the following irregular shapes. (a) (b) (c) 7cm 12m 22m 38cm 8m 7m 9m 21m 17m 24cm 15m 3. Work out the shaded area for each of the following (all measurements are given in centimetres): (a) 15 (b) (c) 5 (d) 3 8

9 4. Calculate the areas for each of the following shapes. (a) 18 (b) (c) (d) (e) (f)

10 Geometr (9) Volume of a prism 1. Without a calculator find (a) the base area (b) the volume for the following cuboids (i) (ii) (iii) 6m (iv) 5m 5cm 9m 4m 7cm 4cm 14m 6m 11m 5m (v) (vii) 7cm (viii) 5cm 15cm 0.5m 19cm 7cm 11m 8m 8cm 32cm 3 2. A classroom has a volume of 780m, if the length and width of the room are 8m and 7.5m respectivel, how high is this classroom? 3. Bricks with dimensions 25cm b 12cm b 9cm are being used to build a wall. 3 (a) Find the volume of one brick (i) in cm (ii) in 3 m. (b) If the wall is to have a total volume of 675 m 3, how man brick will we need? 4. Without a calculator find the volume for each of the following triangular based prisms. (a) (b) 12m (c) 10cm 4cm 5m 7m 6cm 3cm (d) (e) (g) 11cm 6cm 16m 13cm 15cm 5cm 11cm 8m 12cm 17m 24m 5. Workout the volume for each of the following, giving our answers to 2 decimal places.

11 (a) (b) (c) (d) 14cm 3.8m 7cm 7.8cm 4.2m 9cm 2cm (e) (f) (g) (h) 8.9cm 7.5m 6cm 16m 13cm 14cm 6.5m 3m 9cm 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) 8 (e) 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.

12 Geometr (10) Tpes of Polgons 1. Name each of the following tpes 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 A B L K (b) ABCD (c) FGHI (d) CDEF D C J (e) CFIJ (f) DEIJ E F I G H

13 Geometr (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 ou notice. Cube Cuboid Square based pramid Tetrahedron Triangular prism Number of Faces (F) Number of Vertices (V) Number of Edges (E) F + V E 2. A Sweet is of the shape of a triangular prism until someone cuts awa 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 awa the top corner using a rubber and redraw it to look as though someone had cut it awa. (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 tpe drawn in class. State which the could be. [Some will have more than one answer!] (i) (ii) (iii) (iv)

14 5. This is a cuboid (edges not equal in length) Use tracing paper to cop the outline and dotted (hidden) lines into our exercise book. On our diagram draw different planes of smmetr. 6. This is a cube (all edges equal). It will have three planes of smmetr similar to the cuboid in question 5. i) Use tracing paper to cop the outline and dotted lines into our exercise book. Draw a plane of smmetr. ii) Repeat the exercise of (i) as man times as ou need to until all planes of smmetr have been found. iii) How man planes of smmetr does the cube have? 7. This is a square based pramid i) Use tracing paper to cop the outline and dotted lines into our exercise book. Draw a plane of smmetr. ii) Repeat the exercise of (i) as man times as ou can have until all planes of smmetr have been found. 8. This is a sphere with a plane of smmetr. Draw a sphere into our book along with another plane of smmetr. How man planes of smmetr could be drawn?

15 9. This is a clinder. Draw a clinder into our book with a different plane of smmetr. How man planes of smmetr could be drawn? 10. (i) Using a square (side 2cm) complete a net for a square based pramid each edge of which will be length 6 cm. (ii) Draw on card a net for a square based pramid of length 6cm. Add suitable flaps, cut out our 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 our net; Use a pritt stick to glue together in the form of a regular tetrahedron.

16 Geometr (12) Angles in a straight line Work out the lettered angles for each of the following diagrams. Remember to show our working. All diagrams are not drawn to scale b 17 c 53 a d e f m 3m 2n n 2n 42 2r r p p t t u 15. u 4u 4x x k k 2k 49 35

17 Geometr (13) Angles at a point Work out the lettered angles for each of the following diagrams. Remember to show our working. All diagrams are not drawn to scale a 126 b c d 40 d 72 e 2e f 87 3f g 58 4g h 75 h i 2i i k m k m 2m 3n 2n 71 n q q 96 p 3r 2r r 2p 3q 3r

18 Geometr (14) Angles associated with parallel lines Work out the missing angles in each of the following triangles. Remember to show our working. All diagrams are not drawn to scale a c b d e 158 f 18 g h 4. i j p q r k n m 46 s t z u x 104 b 55 a v x p 71 r 37 q 64

19 Geometr (15) Angles in a polgon Work out the missing angles in each of the following triangles. Remember to show our working. All diagrams are not drawn to scale b 58 a c k i 18 2i k+12 m 42 m a c 11 d b g h k i j l e f 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 b x, x + 42 and x What is the value of x?

20 16. Calculate the missing angles for each of the following parallelograms (a) a (b) (c) 68 d e g b c 29 f 127 h i 17. A quadrilateral has three angles of size 45, 132 and 77. What is the size of the fourth angle? m 18. Find the missing angles in the kite drawn opposite n 19. Find the interior angles for each of the following regular polgons (a) A pentagon (b) A nonagon (c) A dodecagon (d) An octagon 20. Find the size of angle x opposite. Give a reason for our answer. x 21. If the two polgons partl drawn below represent two n sided regular polgons joined at one side how man sides does each polgon have? 48

21 Geometr (16) Pthagoras Theorem Answer all the following questions, showing our working. 1. Find x 2. Find the length of PR P 6cm x 5cm 8cm R 12cm Q 3. Find EF correct to 1 decimal place. 4. Find p correct to 2 decimal places E 22cm 10cm 18cm p D 7cm F 5. Find a correct to the nearest whole number 5.4m a 3.6m 6. Find the length of the missing side, giving our answer to a suitable degree of accurac. A C 13m 9m B

22 7. Calculate the length of the diagonal in the rectangle drawn opposite, giving our answer correct to three significant figures. 10cm 16cm D 8. Triangle DEF is isosceles Calculate the lengths of (i) FM (ii) DF. 15cm E M 24cm F 9. The diagram drawn opposite represents a ladder placed against a wall. Calculate the length of the ladder correct to 5.4m the nearest centimetre. 2.7m 10. The dotted line on this map represents the journe 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 journe. {answers to one decimal place}. D B C A km

23 Geometr (17) Pthagoras Theorem II Answer all the following questions, showing our working. 2. Find x 2. Find the length of XY Y 20cm x 26cm 16cm Z 24cm X 3. Find EF correct to 2 decimal places 4. Find x correct to the nearest whole number. E 40m x D 23cm 27m 29cm F 5. Find d correct to one decimal place. 6. Find BC. 4.5cm B 8cm 11cm d C 15cm A 7. The diagram represents the front end of a garden shed. 2.9m Find the width of the shed correct to one decimal place. 2.2m 3.1m

24 8. Calculate the length of a rectangle which has width 8cm and diagonal of length 21cm. Giving our answer to a suitable degree of accurac. 9. Two planes are fling over the village of Colne, one directl above the other when the are picked up b a radar station some 10km awa 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. 15km 13km 10km Colne 10. Calculate the values of x and in the diagram below, giving our answers correct to 2 dp. 8 m 2.4 m x 6.5 m 4.8 m

25 Geometr(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) x x (c) Rotate 90 anticlockwise centre (0, 3) (d) Rotate 90 clockwise centre (4, 0) x x (e) Rotate 180 Centre (3, 2) (f) Rotate 90 clockwise centre (2, 2) x x

26 (g) Rotate 180 centre ( 1,3) (h) Rotate 90 anticlockwise centre (2, 0) x x (i) Rotate 270 clockwise centre (0, 0) (j) Rotate 270 clockwise centre (5, 1) x x Describe the transformation which has taken place in each of the following mappings of triangle A onto the shaded triangle. (a) (b) x x

27 3. Describe the transformation which will map triangle ABC onto triangle PQR. 4 3 B P 2 1 A C x 1 Q R (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.

28 Geometr(19) Transformations : Reflections 11. Reflect each of the following in the given line. (a) Reflect in the line x 1 (b) Reflect in the line x x (c) Reflect in the line 1 (d) Reflect in the line x x x (e) Reflect in the line x 2 (f) Reflect in the line x x

29 (g) Reflect in the line 2 (h) Reflect in the line x x x (i) Reflect in the line x (j) Reflect in the line x x x Describe the transformation which has taken place in each of the following mappings of triangle A onto the shaded triangle. (a) (b) x x

30 13. Describe the transformation which will map triangle ABC onto triangle PQR P A 4 x 2 3 R 4 Q B C (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 2 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 x 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 x 1 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 x 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 x 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 0 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 x 4 and label the image P Q R.

31 Geometr(20) Transformations : Enlargements 1. Enlarge the shape below b a scale factor of three centre of enlargement O. Label the image P. O P 2. Draw shape ABCD after an enlargement with scale factor 2 centre D. Label the image A 1 B 1 C 1 D 1. D A B C 3. Enlarge the triangle LMN b a scale factor 4 centre P. P M L N

32 4. The object L has been enlarged onto Image L. (a) Identif the centre of enlargement and label it C. (b) State the scale factor of the enlargement. L L 5. Obtain the centre and scale factor of the enlargement of the shaded shape drawn below.

33 Geometr(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) x (b) x

34 (c) x (d) x 2. Enlarge LMN b a scale factor of 2 centre ( 1, 1); Label the image L 1 M 1 N L M N x

35 3. Enlarge ABC b a scale factor of 4 centre (1, 2). Label the image A 1 B 1 C 1 5 A B 1 C 0 x Enlarge the object below with centre ( 3, 2) b a scale factor x Enlarge the object b a scale factor of 3 centre of enlargement (4,5) x 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 b a scale factor of 3 centre (1, 3)

36 Geometr(22) 1. The diagram drawn opposite shows four triangles drawn in different positions. Using the vector notation describe the translation which will map (i) (ii) (iii) (iv) (v) ABC onto EDG ABC onto HIJ ABC onto PQR PQR onto EDG HIJ onto PQR Transformations : Translations 2. Using the drawn triangle opposite A C x B i) draw the image A B C after a translation of ii) iii) ABC b 4 2 draw the image A B C after a translation of ABC b 6 2 draw the image A B C after a translation of ABC b 2 3 (iv) Describe the translation which maps A B C onto A B C A B C P x -2 Q R E D H I G J 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). 2 (b) Draw the image of STUV after a translation of. Label the image S T U V (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 (c) Draw the image of L M N after a translation of 4. Label the image L M N Label the image L M N 2

37 Geometr (23) Bearings I 1. Write down the bearings of A from B for each of the following diagrams. (a) (b) N N A B B (c) A N (d) N A A B B (e) N (f) N A B N B (g) (h) N A B B A A

38 2. Write down the bearings each of the following demonstrates (b) (b) N K Y J (c) N G X (d) N C F D (f) N (f) N S V T U (g) N (h) P N N M Q

39 3. For each of the following draw a scale diagram to represent the journe, 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 acht 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 acht 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 awa, while the other is on a bearing of 310 from the lighthouse and 17km awa. (a) Using a scale of 1cm = 2km, draw a scale diagram to represent the information above. (b) How far apart are the ships?

40 Geometr (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 journe. How far is the ship awa from its original position? On what bearing could the ship have originall taken? 3. The insert given shows the towns of Appleton, Barton, Cotle, Dove and Eccles. Using the diagram work out the bearing of (a) Eccles from Appleton (b) Cotle from Dove, (c) Dove from Barton, (d) Appleton from Cotle, (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 identif on the insert the position of the ship.

41 Geometr (24) Inserts Insert 1 Appleton Barton Dove Eccles Cotle Insert 2 P Q

42 Geometr (25) Introduction to Trigonometr Opposite Hpotenuse Remember Adjacent Exercise With each of the right-angled triangles below, write the name of each lettered side f a b e d h g c i j l p n s q k m r t b w u x a v z 10. e d c

43 Geometr (26) The Sine ratio 1. For each of the following triangles below: (i) (ii) (iii) write down the length of the opposite side write down the length of the hpotenuse Opposite calculate the ratio Hpotenuse Give our answer correct to 3 decimal places where necessar. Example 2cm 4cm 30 (i) Opposite = 2cm (ii) Hpotenuse = 4cm (iii) Opposite Hpotenuse 4 a. 2cm b. c. 6cm 2.5cm 12cm 4cm 3cm cm d. e. f. 4cm 2.2cm cm 5cm 4.9cm cm Look on our calculator for a button Sin, 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 Sin and then the buttons 5 0 followed b =

44 2. (a) Find the sine of each of the angles in the triangles of question 1, giving our answer to 3 decimal place. (b) Compare each of our results to our answers to part (iii) in question 1. What do ou notice? (c) Complete the equation opposite Sin x = x Exercise For each of the following triangles, find the length of the lettered side, giving our answers correct to 1 decimal place cm 54 2cm 32 6cm b c a mm 5cm e 71 4m f d i 3cm g 9m 30 h cm 4cm j 52

45 Geometr (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 hpotenuse Adjacent c. calculate the ratio Hpotenuse Give our answer correct to 3 decimal places where necessar. a. b. 3cm c cm 4.5cm cm 2.5cm cm d. 4.5cm e. f cm 8cm cm 5.4cm 4.9cm Look on our calculator for a button ( Cos is short for cosine) Cos, we use this if we want to find the Cosine of the angle 2. (a) Find the sine of each of the angles in the triangles of question 1, giving our answer to 3 decimal place. (b) Compare each of our results to our answers to part (iii) in question 1. What do ou notice? (c) Complete the equation opposite Cos x = x

46 Exercise For each of the following triangles, find the length of the lettered side, giving our answers correct to 2 decimal place cm b 45 8cm 7cm c 18 a mm 5m e d 7.2cm f g 66 6cm h 24 8cm i 29 12cm 10. j m Look on our calculator for a button Tan, 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. 3cm 5cm 2.5cm 6.5cm 4cm cm 22.62

47 Exercise For each of the following triangles, find the length of the lettered side, giving our answers correct to 2 decimal place c b a 31 7cm cm 7m f 3m d e 6m cm g j 65mm 58 5cm i cm 74 8cm h 8

48 Geometr (28) Using The three trig ratios 1. For each of the following questions find the length of the missing sides. Giving our answers correct to one decimal place. (a) (b) (c) 32 4cm x 8cm j 2.4m (d) (e) (f) x x 70 5cm 41 x 4.5cm 58 9cm x C 2. For the triangle drawn opposite find the length of (i) AB (ii) BC 9cm 62 A B R 3. For the triangle drawn opposite find the lengths of 8cm (i) PR (ii) QR Q 52 P 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.

49 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 15cm x Find the height of these stairs correct to one decimal place. h m 11. A kite is fling 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. B 10.2m 75 A C 11.4m 8.5m m

50 Geometr (29) Finding an angle using trigonometr For each of the following find the size of the missing angle x 13cm 4.2cm 5cm x x 25cm 2.4cm 10cm 4. 14cm cm 21cm x 8cm x 7cm 3.7cm x cm x 7cm 27cm x 3cm x 18cm 4.2cm m 3.4m x 7.4cm x 9m 6.5m 10.3cm x

51 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. 50m 150m 20. the diagram represents a lighthouse of height 135mand a bo standing at point P, 375m awa. What is the angle of elevation from the bo to the top of the lighthouse? x 375m

52 Geometr(30) Circle theorems Work out the lettered angles in each of the following diagrams (1) (2) 50 a b 78 (3) (4) 47 c 95 x 15 d 17 (5) (6) e g (7) (8) i h x

53 Geometr(31) Circle theorems Work out the lettered angles in each of the following diagrams b e c 70 f a n 150 m k h s 95 x 28 r t

54 Geometr(32) Circle theorems Work out the lettered angles in each of the following diagrams b 43 a a b d z c x f e 40 d m 38 n q v u p 54

55 Geometr(33) Circle theorems Work out the lettered angles in each of the following diagrams c a d b f h e g i p r q 70 k 21 n m u 95 x 74 s t 110 w

56 Geometr(34) Circle theorems Work out the lettered angles in each of the following diagrams a c 36 b e n m r t s 70 u 30

57 Geometr (35) Circle Theorems Work out the lettered angles in each of the following diagrams (1) (2) 15 a b c (3) (4) 32 d f e (5) g (6) 85 i 40 h (7) (8) n m k 25

58 Geometr (36) The Sine rule 1. Work out the lettered side for each of the following: (a) (b) 7 cm a 30º b 17 cm 40º 95º 20º (c) (d) 6 m 10º 70º c 3.7 cm d 125º 35º (e) 8.2 cm (f) 27º 50º 45º e f 12 cm 68º 2. In triangle STU, ST 7. 5 cm, Work out the length of TU. ˆ TSU 45 and ˆ U TS In triangle LMN, LM 4. 7 cm, LMN ˆ 54 and MLN ˆ 78 Work out the length of MN.

59 4. Work out the lettered angle for each of the following: (a) (b) A 6 cm 70º 13 cm 24 cm 7 cm B 20º (c) C (d) 3 m 115º 6.3 cm 5.4 cm 9 m 88º D (e) 7.6 cm 7.1 cm (f) 35º E 69º 23 cm F 30 cm 5. In Triangle LMN, LM 6cm, Work out the size of LM ˆ N. ˆ 35 M LN and LN 7 cm. 6. In the triangle XYZ when XZ 3. 5cm, YZ 5. 8 cm and Work out the size of ˆ XZY. ˆ X YZ In the triangle STU when ST 8. 3cm, TU 4. 9 Work out the size of ˆ TSU. cm and ˆ U TS 29

60 8. For each of the triangles below find the lengths or angles required. (a) (b) 6.5cm 8cm F E cm 122 Find Angle E Find length F (c) N (d) 10cm 9.7cm 8.1cm 70º cm M Find Angle N Find Angle M

61 Geometr (37) The Cosine rule 9. Work out the lettered side for each of the following: (a) 6 cm a (b) 8.5 cm 75º 9.4 cm 20º 7 cm b (c) 115º 5.7 cm (d) 5 m 7.3 cm 77º d c 13 m (e) e 16 cm (f) 50º 29 cm 28 cm 8 cm 124º f 10. In the triangle LMN, LM 8cm, Work out the length of MN. ˆ 35 M LN and LN 5 cm. 11. IN the triangle XYZ, XY 3. 5cm, YZ 5. 8 cm and Work out the length of XZ. ˆ X YZ 68

62 12. Work out the lettered angle for each of the following: (a) A 7 cm (b) 14 cm 6 cm B 10 cm 8 cm 14 cm (c) (d) 6 m D 15 m 11 cm 8 m 9 m 5 m C 13. In triangle ABC sides are AB = 8cm, AC = 6cm and BC = 4cm. Work out the size of AB ˆ C. 14. In triangle HIG sides are HI = 6.4cm, HG = 7.1cm and IG = 4.5cm. Work out the size of Hˆ IG. 15. In triangle ABC where AB = 5cm, BC = 6cm and AC = 4.3cm. Work out the size of the largest angle.

63 Geometr(38) Area continued 1. Calculate the area for each of the following triangles (a) (b) (c) 9cm 5.7cm cm 8.3cm 60 6cm cm (d) (e) 21cm 17 19cm 34cm 40 57cm 2. (a) Work out the size of angle A in the triangle below B 10cm 20 C A 15cm (b) Hence find the area of triangle ABC 3. (a) Work out the size of angle D in the triangle below. E 11cm 8cm F 7cm D (b) Hence find the area of triangle DEF

64 4. Work out the area of the triangles drawn below a) (b) 25 20cm 20cm 80 17cm 7cm 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 Find the area of the shape drawn below. [Be careful..this involves a lot of previous knowledge] 10cm 10cm 10cm 10cm 10cm

65 Geometr (39) Volume of a Pramid/Sphere 8. For each of the following work out the volume, where appropriate giving our answers to 2 decimal places. (i) (ii) (iii) 9cm 10cm 5cm 7cm 5cm 5cm (iv) (v) 18 m (vi) 8 cm 3.5mm Area = 105 cm 2 15mm 2.2cm 9. The diagram shows the cork top of a bottle with dimensions given. Find its volume. 1.4cm 1cm 5cm cm The diagram is of a garden pot with square base 50cm and top 60cm. Find its volume. 95cm

66 11. Find the volume of the shape drawn below 70 cm 20 cm 12. The volume of a square based pramid with height 12cm is 144 cm 3. Find the length of the side of the square. 13. A cone has volume 108 m 3. Find its radius when its height is 4m.

67 Geometr(40) Surface Area 1. Calculate the surface area for each of the following shapes (a) (b) (c) 4cm 7cm 20cm 15cm 12cm 5cm 8cm (d) (e) (f) 24cm 40cm 9cm 40cm 25cm 2. Work out the total surface area of the hemisphere drawn below. 10 cm 3. Calculate the surface area of the shape drawn below. (a) (b) 3mm 15mm 18mm 10mm 3mm 5mm 4mm

68 Geometr (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) 5 cm a 9 cm 27 cm (b) b 9cm 6 cm 27 (c) c 8 cm 6 4cm d 7 cm 3cm D B 2. Show that triangle ABC is similar to triangle ADE 9cm Hence work out the length of (i) DE 7cm (ii) CE A 10cm C E 3. Given that the two rectangles drawn are 16 similar find the height of the rectangle 2 8 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

69 Geometr (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) 5 cm 2 A 9 cm 27 cm b) B 32 cm 2 8 cm 20 cm c) 45 cm 2 C 15 cm 6 cm d) D 15 cm 45 cm 5 cm 2. A triangle has sides 5cm, 12cm and 13cm, and has an area of 30cm 2. A similar triangle has an area of 120cm 2. Find the lengths of each side of the larger triangle. 3. Two similar cones have heights 4cm and 8cm respectivel. If the volume of the larger cone is 56cm 3, find the volume of the smaller cone.

70 4. Two similar spheres have masses of 24kg and 648kg respectivel. If the radius of the smaller sphere is 5cm find the radius of the larger sphere. 5. In the diagram below the two clinders are similar. Find the length of the lettered side. 3cm 3 3cm 192cm 3 x cm 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 50cm 2, find the area of the trapezium ABZY. X A B Y Z 7. Find the volume of the larger solid of the two drawn below, given that both solids are similar. 24cm 3 8cm 12cm 8. Two similar solids have surface areas 20m 2 and 45m 2 respectivel, 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, respectivel. Given that the surface are of the larger sphere is 75cm 2, find the surface are of the smaller sphere.

71 Geometr(43) Vectors 1. Write the components of each vector in the diagram below. a b c d 2. Write down in component form each of the following vectors H C F A B D 3. B 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)? E 5. Draw suitable diagrams to illustrate each of the following vectors. Label each vector accordingl a) a b) b c) AB d) LM Given a and b work out the vector a b. Represent our answer on a 2 3 suitable diagram. 7. Find the values of the missing letters in each of the following additions. G a 5 3 a) 1 b e b) 7 d 5 1 m 8 c) n 4 3

72 8. Use the diagram given to find the appropriate component form for the vector equivalent to a. x b. x z c. x z a x z Given a and b work out the vectors 0 3 a) a 2b b) a b c) 2a 3b 1 d) a 2b 2 a

73 Geometr(44) Vector Geometr 1. Given the vectors a and b below draw diagrams to represent each of the following vectors a b a) a b b) a b c) 2 a b d) a 2b e) 2b 3a 2. In the parallelogram ABCD drawn opposite E B C and F are the midpoints of AB and CD respectivel. If AD a and AE b, write in terms of b E F a and b A (i) AB (ii) AF (iii) AC (iv) BD a D N 3. In the triangle LMN points P and Q are the midpoints of the lines LN and MN respectivel. Given that LN a and LM b m write in terms P Q of a and b (i) LP (ii) MN (iii) NQ (iv) LQ L M

74 Z 4. In the triangle XYZ the point T is such that YT=3ZT. Given that and q XZ p and XY q, express in terms of p T (i) YZ (ii) YT (iii) XT X Y 5. The diagram below consists of three equilateral triangles joined together. A B a O d D C Work out each of the following vectors a) AD (b) AB (c) OB (d) AC 6. OABCDE is a regular hexagon with OA represented b the vector a and OE represented b the vector e. Find the vectors representing (i) AB (ii) OC (iii) AD

75 Geometr(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 A O P Find in terms of a and b B (i) AB (ii) AP (iii) OP O A 2. OACB is a square with OA a and OB b P is a point on AC such that AP : PC = 1 : 3 and Q P is on OB such that OQ : QB = 3 : 1. Find in terms of a and b Q (i) OQ (ii) OP B C M 3. OLMN represents a kite with OL a, ON b and LM c P Q Points P, Q, R and S are the midpoints of the lines LM, MN, ON and OL respectivel. L N a) Find in terms of a, b and c (i) NM (ii) SR (iii) PQ S R b) Comment on our finding in part (a) O 4. OABC is a parallelogram with OA a and A B OB c P is a point on AC such that AP PC 1 3 and a P Q Q is the midpoint of BC. Find in terms of a and b O c C (i) OP (ii) OQ

76

77 Geometr (46) Special Curves 1. (a) Cop and complete the table below for the graph of sin x x (b) On Graph paper draw the graph of sin x (c) Use our graph to solve each of the following equations (i) sin 75 (ii) sin x 0. 8 (iii) sin x (a) Cop and complete the table below for the graph of cos x x (b) On Graph paper draw the graph of cos x (c) Use our graph to solve each of the following equations (i) cos x 0. 6 (ii) cos x On the calculator there is a button e x, meaning exponential of x. a) Use this button to complete the table below. x b) On graph paper draw the graph of x e 4. Given that sin state another angle which would give the answer Given that cos state another angle which would have given the answer

78 Geometr (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

79 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) respectivel. 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

80 Geometr (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) (ii) Work out the lengths of FB and FC What is the length of the diagonal in the rectangle ABCD? (iii) Find the angle AB ˆ F (iv) Work out the angle made between FC and the plane ABCD. 3. The diagram opposite is of a square based pramid with side 7cm and slanted edge 9cm. Work out a) The length of the diagonal AC b) The height of the Pramid, EF c) The angle BA ˆ E 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 Work out A BF ˆ 20 a) The height AF b) The length of the diagonal BE c) The angle EB ˆ D

81 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. 4m. The base ABCD forms a rectangle with dimensions 12m b ABF and DCE are identical isosceles triangles with slanted edge 6m. G and H are the midpoints of AB and DC respectivel. Work out (a) The lengths FG and EA (b) The perpendicular length from point F to the base ABCD (c) The angle (d) Angle ED ˆ A FG ˆ H 7. The picture is of one of the largest pramids in Egpt, the pramid of Giza. As one of the oldest seven wonders of the world its height was approximatel 146 m tall and the square base is approximatel 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.

82 Geometr (49) The General Triangle 1. Work out the lettered side for each of the following: (a) (b) 7 cm a 30º b 17 cm 40º 95º 20º (c) (d) 6 m 10º 70º c 3.7 cm d 125º 35º 2. In triangle STU, ST 7. 5 cm, Work out the length of TU. ˆ TSU 45 and ˆ U TS In triangle LMN, LM 4. 7 cm, LMN ˆ 54 and MLN ˆ 78 Work out the length of MN. 4. Work out the lettered angle for each of the following: (a) A (b) 6 cm 70º 13 cm 24 cm 7 cm B 20º

83 (c) C (d) 3 m 115º 6.3 cm 5.4 cm 9 m 88º D 5. In Triangle LMN, LM 6cm, Work out the size of LM ˆ N. ˆ 35 M LN and LN 7 cm. 6. In the triangle XYZ when XZ 3. 5cm, YZ 5. 8 cm and Work out the size of ˆ XZY. ˆ X YZ In the triangle STU when ST 8. 3cm, TU 4. 9 Work out the size of ˆ TSU. cm and ˆ U TS For each of the triangles below find the lengths or angles required. (a) (b) 6.5cm 8cm F E cm 122 Find Angle E Find length F 9. Work out the lettered side for each of the following: (a) 6 cm a (b) 8.5 cm 75º 9.4 cm 20º 7 cm b

84 (c) 115º 5.7 cm (d) 5 m 7.3 cm 77º d c 13 m 10. In the triangle LMN, LM 8cm, Work out the length of MN. ˆ 35 M LN and LN 5 cm. 11. IN the triangle XYZ, XY 3. 5cm, YZ 5. 8 cm and Work out the length of XZ. ˆ X YZ Work out the lettered angle for each of the following: (a) A 7 cm (b) 14 cm 6 cm B 10 cm 8 cm 14 cm (c) (d) 6 m D 15 m 11 cm 8 m 9 m 5 m C

85 13. In triangle ABC sides are AB = 8cm, AC = 6cm and BC = 4cm. Work out the size of AB ˆ C. 14. In triangle HIG sides are HI = 6.4cm, HG = 7.1cm and IG = 4.5cm. Work out the size of Hˆ IG. 15. In triangle ABC where AB = 5cm, BC = 6cm and AC = 4.3cm. Work out the size of the largest angle.