Mathematics. Geometry. Stage 6. S J Cooper

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1 Mathematics Geometry Stage 6 S J Cooper

2 Geometry (1) Pythagoras Theorem nswer all the following questions, showing your 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 your answer to a suitable degree of accuracy. C 13m 9m

3 7. Calculate the length of the diagonal in the rectangle drawn opposite, giving your 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 5.4m to the nearest centimetre. 2.7m 10. The dotted line on this map represents the journey of a ship travelling from to D stopping at two ports on route at and C. Calculate the total length of this ships journey. {answers to one decimal place}. D C km

4 Geometry (2) Pythagoras Theorem II nswer all the following questions, showing your 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 C. 4.5cm 8cm 11cm d C 15cm 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

5 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. 15km 13km 10km Colne 10. Calculate the values of x and y in the diagram below, giving your answers correct to 2 dp. 8 m 2.4 m x 6.5 m y 4.8 m

6 Geometry (3) Introduction to Trigonometry Opposite Hypotenuse Remember djacent 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 y z 10. e d c

7 Geometry (4) 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 hypotenuse Opposite calculate the ratio Hypotenuse Give your answer correct to 3 decimal places where necessary. Example 2cm 4cm 30 (i) Opposite = 2cm (ii) Hypotenuse = 4cm (iii) Opposite Hypotenuse 4 a. 2cm b. c. 6cm 2.5cm 12cm cm cm cm d. e. f. 4cm 2.2cm cm 5cm 4.9cm cm Look on your 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 by =

8 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 Sin x = x Exercise For each of the following triangles, find the length of the lettered side, giving your answers correct to 1 decimal place cm 54 2cm 32 6cm c b a mm 5cm e 71 4m f d cm g 9m 30 h i 10. 6cm 4cm j 52

9 Geometry (5) 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 djacent c. calculate the ratio Hypotenuse Give your answer correct to 3 decimal places where necessary. 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 your 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 cosine 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 Cos x = x

10 Exercise For each of the following triangles, find the length of the lettered side, giving your answers correct to 2 decimal place cm b 45 8cm 7cm c 18 a d 18 6mm 5m 32 e 7.2cm f g 66 6cm h 24 8cm i 29 12cm 10. j m Look on your 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

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

12 Geometry (6) 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) 32 4cm x 8cm j 2.4m (d) (e) x (f) x 70 5cm 41 x 4.5cm 58 9cm x C 2. For the triangle drawn opposite find the length of (i) (ii) C 9cm 62 R 3. For the triangle drawn opposite find the lengths of 8cm (i) PR (ii) QR Q 52 P 4. In triangle C, ngle C = 90, ngle = 74 and C = 19cm. Find the length of C. 5. In triangle PQR, ngle Q = 90, ngle P = 37 and PR = 3.2cm. Find the length of PQ.

13 6. In triangle LMN, ngle M = 90, ngle L = 81 and LN = 14cm. Find the length of MN. 7. In triangle DEF, ngle F = 90, ngle D = 21 and DF = 3.2cm. Find the length of EF. 8. In an isosceles triangle C, angle is 47 and the length = C = 10cm. Calculate the length of C. 9. For the diagram drawn below find the lengths of x and y 15cm x 20 y Find the height of these stairs correct to one decimal place. h m 11. 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, and C from the ground. 10.2m 75 C 11.4m 8.5m m

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

13. In triangle C, angle = 90, C = 60 cm and C = 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.

15 13. In triangle C, angle = 90, C = 60 cm and C = 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. n 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 boy standing at point P, 375m away. What is the angle of elevation from the boy to the top of the lighthouse? x 375m

16 Geometry(8) 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)

(g) Rotate 180 centre ( 1,3) (h) Rotate 90

Describe the transformation which has taken

17 (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 onto the shaded triangle. (a) (b)

18 3. Describe the transformation which will map triangle C onto triangle PQR. 4 3 y P C x 1 Q 2 R (a) Plot the points (1, 3), (4, 2) and C(4, 5) and join up the points to form a triangle. (b) Rotate triangle C through 90 anticlockwise, centre (1, 1) and label the image 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 ( 1, 3), ( 4, 2) and C( 4, 1) and join up the points to form a triangle. (b) Rotate triangle C through 90 clockwise, centre ( 2, 0) and label the image 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.

19 Geometry(9) Transformations : Reflections 11. Reflect each of the following in the given line. (a) Reflect in the line x 1 (b) Reflect in the line y 1 (c) Reflect in the line y 1 (d) Reflect in the line x 2 (e) Reflect in the line x 2 (f) Reflect in the line y 0

20 (g) Reflect in the line y 2 (h) Reflect in the line y x (i) Reflect in the line y x (j) Reflect in the line y x 12. Describe the transformation which has taken place in each of the following mappings of triangle onto the shaded triangle. (a) (b)

21 13. Describe the transformation which will map triangle C onto triangle PQR. y P x 2 3 R 4 Q C (a) Plot the points (1, 3), (4, 2) and C(4, 5) and join up the points to form a triangle. (b) Reflect triangle C in the line y 2 and label the image 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 y 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 y 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 y x and label the image S T U. 19. (a) Plot the points ( 1, 3), ( 4, 2) and C( 4, 1) and join up the points to form a triangle. (b) Reflect triangle C in the line y 0 and label the image 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

For each of the following state (i) the centre of enlargement (a) 9

22 Geometry(10) 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 (a) y (ii) the scale factor of the enlargement x (b)

23 (c) (d) y x 2. Enlarge LMN by a scale factor of 2 centre ( 1, 1); Label the image L1M1N1

24 3. Enlarge C by a scale factor of 4 centre (1, 2). Label the image 1 1 C 1 y C 0 x Enlarge the object below with centre ( 3, 2) by a scale factor 3. y x Enlarge the object by a scale factor of 3 centre of enlargement (4,5) y x 6. (a) Plot the points (1, 2), ( 3, 2) and C(3, 0) and join up the points to form a triangle C. (b) Enlarge the triangle C by a scale factor of 3 centre (1, 3)

25 Geometry(11) 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) C onto EDG (ii) C onto HIJ (iii) C onto PQR (iv) PQR onto EDG (v) HIJ onto PQR P Q R C E D H I G J 2. Using the drawn triangle opposite C i) draw the image C after a translation of ii) iii) C by 4 2 draw the image C after a translation of C by 6 2 draw the image C after a translation of C by 2 3 (iv) Describe the translation which maps C onto 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). 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

26 Geometry(12) Circle theorems Work out the lettered angles in each of the following diagrams (1) (2) 50 a b 78 (3) (4) d c 95 x 17 (5) (6) 60 e g (7) (8) i y h x

27 Geometry(13) 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 y t

28 Geometry(14) Circle theorems Work out the lettered angles in each of the following diagrams b 43 a a b d z y c x f e 40 d m 38 n q v u p 54

29 Geometry(15) Circle theorems Work out the lettered angles in each of the following diagrams a c d b f h e g i p r q 70 k 21 n m u 95 x 74 s t 110 y w

30 Geometry(16) 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

31 Geometry (17) 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 25 m k

32 Geometry (18) 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) 9cm 10cm 5cm 7cm 5cm 5cm (iv) (v) 18 m (vi) 8 cm 3.5mm rea = 105 cm 2 15mm 2.2cm 2. The diagram shows the cork top of a bottle with dimensions given. Find its volume. 1.4cm 1cm 5cm 3. 55cm The diagram is of a garden pot with square base 50cm and top 60cm. Find its volume. 95cm

33 4. Find the volume of the shape drawn below 70 cm 20 cm 5. The volume of a square based pyramid with height 12cm is 144 cm 3. Find the length of the side of the square. 6. cone has volume 108 m 3. Find its radius when its height is 4m.

34 Geometry(19) Surface rea 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

35 Geometry (20) 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 2. Show that triangle C is similar to triangle DE 9cm Hence work out the length of (i) DE 7cm (ii) CE 10cm C E 3. Given that the two rectangles drawn are 16 similar find the height of the rectangle 2 8 labelled. Hence find the areas of the rectangles and. Deduce the relationship between the areas of and and the length ratios of and

36 Geometry (21) 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 9 cm 27 cm 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. triangle has sides 5cm, 12cm and 13cm, and has an area of 30cm 2. similar triangle has an area of 120cm 2. Find the lengths of each side of the larger triangle.

37 3. Two similar cones have heights 4cm and 8cm respectively. If the volume of the larger cone is 56cm 3, 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. In the diagram below the two cylinders are similar. Find the length of the lettered side. 3cm 3 3cm 192cm 3 x cm 6. In triangle XYZ a line parallel to YZ is drawn such that X = 2cm. Given that Y = 3cm and the area of triangle XYZ is 50cm 2, find the area of the trapezium ZY. X 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 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 75cm 2, find the surface are of the smaller sphere.

38 Geometry (22) 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 f 12 cm e 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.

39 4. Work out the lettered angle for each of the following: (a) (b) 7 cm 70º 13 cm 24 cm 6 cm 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, MN 6 cm, Work out the size of LM ˆ N. ˆ 35 LN M LN and 7 cm. 6. In the triangle XYZ when XZ 3. 5cm, YZ 2. 8cm and Work out the size of ˆ XZY. ˆ X YZ In the triangle STU when ST 8. 3cm, TU 4. 9 cm and Work out the size of UTS. U ST ˆ 29

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

41 Geometry (23) The Cosine rule 1. 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 2. In the triangle LMN, LM 8cm, Work out the length of MN. LN ˆ 35 LN M and 5 cm. 3. IN the triangle XYZ, XY 3. 5cm, YZ 5. 8 cm and Work out the length of XZ. ˆ X YZ 68

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

43 Geometry(24) rea 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 in the triangle below 10cm 20 C 15cm (b) Hence find the area of triangle C 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

44 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. [e careful..this involves a lot of previous knowledge] 10cm 10cm 10cm 10cm 10cm

45 Geometry(25) 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 D 3. y drawing a suitable diagram or otherwise state the vector which joins the points (1, 2) and (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 accordingly a) a b) b c) d) LM Given a and b work out the vector a b. Represent your answer on 2 3 a 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

46 8. Use the diagram given to find the appropriate component form for the vector equivalent to a. y b. c. x y z a x z x y x y z a 9. Given a) b) 4 a 0 a 2b a b and 2 b 3 work out the vectors c) 2a 3b d) 1 a 2b 2

47 Geometry(26) Vector Geometry 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) 2a b d) a 2b e) 2b 3a 2. In the parallelogram CD drawn opposite E C and F are the midpoints of and CD respectively. If D a and E b, write in terms of b E F a and b (i) (ii) F (iii) C (iv) D a D N 3. In the triangle LMN points P and Q are the midpoints of the lines LN and MN respectively. 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

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

49 Geometry(27) Vectors concluded 1. Relative to O the position vectors of and are a and b. Point P is a point on such that P = 2P O P Find in terms of a and b (i) (ii) P (iii) OP O 2. OC is a square with O a and O b P is a point on C such that P : PC = 1 : 3 and Q P is on O such that OQ : Q = 3 : 1. Find in terms of a and b Q (i) OQ (ii) OP 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 respectively. L N a) Find in terms of a, b and c (i) NM (ii) SR (iii) PQ S R b) Comment on your finding in part (a) 4. OC is a parallelogram with O a and O O c P is a point on C such that P PC 1 3 and a P Q Q is the midpoint of C. Find in terms of a and b O c C (i) OP (ii) OQ

50 Geometry (28) Special Curves 1. (a) Copy and complete the table below for the graph of y sin x x y (b) On Graph paper draw the graph of y sin x (c) Use your graph to solve each of the following equations (i) y sin 75 (ii) sin x 0. 8 (iii) sin x (a) Copy and complete the table below for the graph of y cos x x y (b) On Graph paper draw the graph of y cos x (c) Use your 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 y b) On graph paper draw the graph of x y 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

The diagram shows a triangular prism with = 8cm, F = 6cm and C = 15cm. (i) (ii) Work out the lengths of F and FC What is the length of the diagonal F D in the rectangle CD?

51 Geometry (29) 3D problems 1. The diagram below represents a cuboid with dimensions = 12 cm, C = 6cm and H = 5cm. H E G F (a) Work out the lengths G and F (b) Work out the angle G makes with the line (c) Work out the angle F makes with the plane D C CD. E 2. The diagram shows a triangular prism with = 8cm, F = 6cm and C = 15cm. (i) (ii) Work out the lengths of F and FC What is the length of the diagonal F D in the rectangle CD? (iii) Find the angle ˆ F (iv) Work out the angle made between C FC and the plane CD. E 3. The diagram opposite is of a square based pyramid with side 7cm and slanted edge 9cm. Work out D a) The length of the diagonal C b) The height of the Pyramid, EF F C c) The angle ˆ E d) The angle E makes with the base CD 4. The diagram is of a wedge used for keeping a door open. The base is square with = 12cm. E ngle Work out F ˆ 20 a) The height F F D b) The length of the diagonal E c) The angle E ˆ D C

5. CDEFGH is a cuboid with dimensions 5cm, 6cm, 14cm as shown. Calculate the size of angle EG. C 5cm D H G 6cm E 14cm F 6. CDEF represents the roof of a building. E 8m F 4m.

Work out (a) The lengths FG and E (b) The perpendicular length from point F to the base CD (c) The angle FG ˆ H (d) ngle ED ˆ 7.

52 5. CDEFGH is a cuboid with dimensions 5cm, 6cm, 14cm as shown. Calculate the size of angle EG. C 5cm D H G 6cm E 14cm F 6. CDEF represents the roof of a building. E 8m F 4m. The base CD forms a rectangle with dimensions 12m by F and DCE are identical isosceles triangles with slanted D H C 12m G 4m edge 6m. G and H are the midpoints of and DC respectively. Work out (a) The lengths FG and E (b) The perpendicular length from point F to the base CD (c) The angle FG ˆ H (d) ngle ED ˆ 7. The picture is of one of the largest pyramids in Egypt, the pyramid of Giza. s 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.

THOMAS WHITHAM SIXTH FORM

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 Geometr (3) Constructions of triangles with protractor REMEMBER DO NOT REMOVE ANY CONSTRUCTION