Question Bank | Unit 5A - Trigonometry - Trigonometrical Ratios, Solutions of Triangles

1. In a triangle $A B C$ , $A C = C B$ and $∠ A C B = 40^{°}$ . Calculate $∠ B A C$ . The point $M$ lies on $A C$ such that $B M = M C$ . Calculate $∠ B M C$ .

2. In the diagram, $P Q = 25 cm$ , $P R = 24 cm$ and $∠ P R Q = 90^{°}$ .

(a) Calculate $Q R$ .

(b) Write down the value of $\cos ∠ Q P R$ , giving your answer as a decimal.

3. In the diagram, $X Y = 20 cm$ , $∠ X = 90^{°}$ and $∠ Y = 60^{°}$ .

Using as much of the information given below as is necessary, calculate

(a) $X Z$ ,

(b) $Y Z$ .

$[\sin 60^{°} = 0 \cdot 866, \cos 60^{°} = 0 \cdot5, \tan 60^{°} = 1 \cdot73]$

4. In the diagram, $∠ A B C = 90^{°}$ and $A B = 25 cm$ .

Using as much of the information below as necessary, calculate $B C$ .

$[\sin ∠ B A C = \frac{12}{13}, \cos ∠ B A C = \frac{5}{13}, \tan ∠ B A C = \frac{12}{5}]$

5. Given that $D B C$ is a straight line, use the information in the diagram to write down the value of

(a) $\sin θ^{°}$ ,

(b) $\cos θ^{°}$ .

6. The vertices of triangle $A B C$ are $A (1, 1)$ , $B (5, 1)$ and $C (7, 9)$ . Calculate

(a) the area of triangle $A B C$ ,

(b) the value of $\tan B \overset{\land}{A} C$

(c) the length of side $A C$ .

7. In the diagram, $A B = 9 cm$ , $A C = 6 \cdot4 cm$ , $A N = 3 \cdot6 cm$ , $\overset{\land}{D} A N = 34^{°}$ and $\overset{\land}{A} N C = 90^{°}$ . Calculate

(a) $C N$ ,

(b) $\overset{\land}{A B N}$

(c) $A D$ .

8. Using the information given in this triangle, write down, but do not solve, an equation which can be used to find $x$ .

(b) Use as much of the information below as is necessary to evaluate

(i) $\sin 140^{°}$ ,

(ii) $\cos 130^{°}$ .

$[\sin 40^{°} = \cos 50^{°} = 0·6428, \cos 40^{°} = \sin 50^{°} = 0·7660]$

9. The sides of a triangle are of length $4 cm$ , $5 cm$ and $6 cm$ . Calculate the smallest angle in the triangle, giving your answer in degrees and minutes.

10. In the diagram, $A \overset{\land}{C} E = 90^{°}$ and $A B = B C = C D = D E = 2 cm$ .

Calculate

(a) $B E$ ,

(b) $D \overset{\land}{A} C$ ,

(c) $B \overset{\land}{X} D$ .

11. Solve the equation $\cos^{2} x = - \cos 35^{°}$ , given that $0 \leq x \leq 180$ .

12. In the diagram, $B C = 4 cm$ , $A \overset{\land}{C} B = 90^{°}$ and $A \overset{\land}{B} D = D \overset{\land}{B} C = 30^{°}$ .

Using as much of the information below as is necessary, calculate

(a) $C D$ ,

(b) $A B$ .

$[\sin 30^{°} = 0 \cdot5, \cos 30^{°} = 0 \cdot866, \tan 30^{°} = 0 \cdot577]$

13. $A B C D$ is a quadrilateral in which $A B = A C = 8 cm$ , $C D = 6 cm$ , $B \overset{\land}{A} C = 34^{°}$ and $C \overset{\land}{A} D = 42^{°}$ . Calculate

(a) $B C$ ,

(b) $A \overset{\land}{D} C$ , giving your answer in degrees and minutes, instructively,

(c) the shortest distance from $B$ to $A D$ .

14. (a) Given that $\sin x^{°} = \sin 47^{°}$ and that $90^{°} < x < 180^{°}$ , write down the value of $x$ .

(b) $A B C$ is a triangle in which $A B = 5 cm$ , $A C = 8 cm$ and $B \overset{\land}{A} C = 20^{°}$ . Using as much of the information given below as is necessary, calculate the area of $Δ A B C$ .

$[\sin 20^{°} = 0·3420, \cos 20^{°} = 0·9397, \tan 20^{°} = 0·3640]$

15. (a) Express ${40·3}^{°}$ in degrees and minutes.

(b) $A B C$ is a triangle in which $B C = 3·5 cm$ . Using as much of the information given below as is necessary, calculate $A B$ .

$[\sin C \overset{\land}{A} B = 0·5, \sin A \overset{\land}{B} C = 0·9954, \sin B \overset{\land}{C} A = 0·9100]$

16. $A B C$ is a triangle in which $B C = 1 cm$ , $A C = 5 cm$ , $A B = x cm$ and $B \overset{\land}{C} A = 127^{°}$ . The side $B C$ is produced to $D$ . Given that $\cos 127^{°} = - 0·6$ ,

(a) state the value of $\cos A \overset{\land}{C} D$ ,

(b) calculate the value of $x^{2}$ .

17. $A B C$ is a triangle in which $A B = 5 cm$ , $A \overset{\land}{B} C = 90^{°}$ and $B \overset{\land}{A} C = 47^{°}$ . The point $D$ on $A C$ is such that $A \overset{\land}{D} B = 90^{°}$ .

Using as much of the information given below as is necessary, calculate

(a) $B C$ ,

(b) $A D$ .

$[\sin 47^{°} = 0·7314, \cos 47^{°} = 0·6820, \tan 47^{°} = 1·0724]$

18. $A B C$ is a triangle in which $A B = 3 cm$ , $A C = 5 cm$ and $∠ B A C = 120^{°}$ . Given that $\cos 120^{°} = - 0·5$ , calculate the length of $B C$ .

19. $L M N$ is a triangle in which $L M = 4.2 cm$ , $M N = 2.8 cm$ and $L \overset{∠}{N} M = 30^{°}$ .

(a) Using the information given above, complete the equation in the answer space.

$(\frac{\sin \overset{∠}{L}}{.......} = \frac{.......}{.......})$

(b) Given that $\sin 30^{°} = 0.5$ , find the numerical value of $\sin \overset{∠}{L}$ .

20. In the diagram, $S Q = 2·8 cm$ , $S R = 8·3 cm$ , $S \overset{\land}{P} Q = 33^{°}$ and $S Q$ is perpendicular to $P R$ . Calculate

(a) $Q \overset{\land}{S} R$ , giving your answer in degrees and minutes correct to the nearest minute,

(b) $S P$ .

21. (a) Evaluate $\cos 115^{°}$ , using as much of the information below as is necessary.

(b) In triangle $P Q R$ , $\overset{\land}{Q} = 90^{°}$ , $\overset{\land}{P} = 65^{°}$ and $P Q = 4 cm$ . Calculate $Q R$ , using as much of the information below as is necessary.

	$25^{°}$	$65^{°}$
sin	0·4226	0·9063
cos	0·9063	0·4226
tan	0·4663	2·145

22. In the diagram, $G P = 1 cm$ , $P H = 2 cm$ , $H Q = 1 cm$ , $Q K = 3 cm$ and $\overset{\land}{H} = 90^{°}$ . Calculate the area of the quadrilateral $G P Q K$ .

23. The sides of a triangle are of length $7 cm$ , $8 cm$ and $9 cm$ . Calculate, as a fraction in its simplest form, the cosine of the angle opposite the $7 cm$ side.

24. In the triangle $A B C$ , $A B = B C$ . The point $D$ on $A C$ is such that $A D = D B$ and $B \overset{\land}{D} C = 74^{°}$ . Calculate

(a) $D \overset{\land}{A} B$ ,

(b) $D \overset{\land}{B} C$ .

25. The angles $A$ , $B$ and $C$ of a triangle $A B C$ are $45^{°}$ , $54^{°}$ and $81^{°}$ respectively, $B C$ , the shortest side of the triangle, is $12 cm$ long. Calculate

(a) the length of the longest side of the triangle,

(b) the length of the shortest perpendicular height of the triangle.

The perpendicular bisector of $B C$ meets $B C$ at $M$ and $B A$ at $N$ . Calculate

(c) $M N$ ,

(d) $C N$ .

26. In the diagram, $B D = 6 cm$ , $A D = 5 cm$ and $B D$ is perpendicular to $A C$ .

(a) Calculate the area of $Δ A B D$ .

(b) Using as much of the information below as is necessary, calculate $A C$ .

$[\sin B \overset{\land}{C} D = \frac{3}{5}, \cos B \overset{\land}{C} D = \frac{4}{5}, \tan B \overset{\land}{C} D = \frac{3}{4}]$

27. In the triangle $D E F$ , $D E = 5 middot;6 cm$ , $\overset{\land}{F} = 90^{°}$ and $\overset{\land}{D} = 18^{°}$ . Calculate the length of $D F$ .

28. In the triangle $P Q R$ , $P Q = 15 cm$ , $P R = 12 cm$ and $\overset{\land}{P} = 65^{°} 30^{'}$ . Calculate the area of $Δ P Q R$ .

29. In triangle $A B C$ , $\overset{\land}{B} = 90^{°}$ , $B C = 3 cm$ , $A C = \sqrt{50} cm$ and $A B = x cm$ .

(a) Calculate the exact value of $x^{2}$ .

(b) In the diagram, $Q R S$ is a straight line, $Q R = 5 \cdot 6 cm$ , $R S = 2 \cdot4 cm$ and the area of $Δ P Q R$ is $10 \cdot5 {cm}^{2}$ . Calculate the area of $Δ P R S$ .

30. In the diagram, $P Q R$ is a straight line and $Q C$ and $R D$ are each perpendicular to the straight line $P C D$ . It is given that $R \overset{\land}{P} D = {19·8}^{°}$ , $P R = 500 m$ and $P C = 150 m$ . Using as much of the information given below as is necessary, calculate

(a) $R D$ ,

(b) $Q C$ .

$[\sin {19·8}^{°} = 0·3387, \cos {19·8}^{°} = 0·9409, \tan {19·8}^{°} = 0·3600]$

31. (a) In the figure, $P R S$ is a straight line, $P Q = 7 cm$ , $P R = 4 cm$ and $\sin Q \overset{\land}{R} S = \frac{3}{8}$ . Calculate the value of $\sin P \overset{\land}{Q} R$ , giving your answer as a fraction in its lowest terms.

(b) In the triangle $X Y Z$ , $X Y = 12 cm$ , $Y Z = 8 cm$ and $Z X = 6 cm$ . Calculate the value of $\cos ∠ X Z Y$ , giving your answer as a fraction in its lowest terms.

32. $A B C D$ is a trapezium in which $A B = 4 cm$ , $A D = 7 cm$ and the area of the trapezium is $49 {cm}^{2}$ , calculate

(a) $A \overset{\land}{B} D$ ,

(b) $D C$ .

33. In each of the following triangles, $O P = 10 cm$ and $O A = 8 cm$ .

(a) If $P \overset{\land}{O} A = 90^{°}$ , calculate the length of $A P$ .

(b) If $O \overset{\land}{A} P = 90^{°}$ , calculate the angle $A \overset{\land}{P} O$ .

(c) If $P \overset{\land}{O} A = 110^{°}$ , calculate the length of $A P$ .

(d) If $A \overset{\land}{O} P$ is an acute angle and the circle, centre $O$ , of radius $10 cm$ is drawn, calculate the angle $A \overset{\land}{O} P$ such that area of Δ OAP = $\frac{area of circle}{5 π}$ .

34. In the triangle $A B C$ , $B C = 10 cm$ , $A C = 8 cm$ , $B \overset{\land}{A} C = 90^{°}$ and $X$ is the foot of the perpendicular from $A$ onto $B C$ . Calculate

(a) $A B$ ,

(b) the area of $Δ A B C$ ,

(c) $A X$ .

35. In the diagram, $A D$ is parallel to $B C$ , $B \overset{\land}{A} D = B \overset{\land}{D} C = 90^{°}$ , $B \overset{\land}{D} A = 30^{°}$ and $B D = 7 cm$ .

Using as much of the information given below as is necessary, calculate

(a) $A D$ ,

(b) $C D$ .

$[\sin 30^{°} = 0·5, \cos 30^{°} = 0·8660, \tan 30^{°} = 0·5774]$

36. In the triangle $P Q R$ , $P Q = 7 cm$ , $Q R = 6 cm$ and $R P = 4 cm$ .

Calculate the value of $\cos P \overset{\land}{Q} R$ , giving your answer as a fraction in its lowest terms.

37. In the diagram, $D \overset{\land}{A} C = A \overset{\land}{C} B = 90^{°}$ , $A \overset{\land}{C} D = {61·4}^{°}$ , $A C = 7·5 cm$ and $B C = 3 cm$ .

Calculate

(a) $A \overset{\land}{B} C$ ,

(b) $C D$ ,

(c) Given that $X$ is the point on $C D$ such that $C X = 8 cm$ , calculate the area of triangle $A C X$ .

38. In the diagram, $Q R S$ is a straight line, $Q \overset{\land}{P} R = 69^{°}$ , $P \overset{\land}{R} Q = 75^{°}$ , $P R = 4 cm$ and $R S = 7·5 cm$ .

Calculate

(a) $P Q$ ,

(b) $P S$ .

39. The triangle $A B C$ is isosceles with $A B = A C$ . The bisector of $A \overset{\land}{B} C$ meets $A C$ at $D$ and the bisector of $B \overset{\land}{A} C$ meets $B D$ at $X$ . Given that $A \overset{\land}{B} C = 80^{°}$ , calculate

(a) $B \overset{\land}{A} C$ ,

(b) $B \overset{\land}{D} C$ ,

(c) $A \overset{\land}{X} B$ .

40. Using as much of the information below as is necessary, evaluate

(a) $\sin 110^{°}$ ,

(b) $\cos 130^{°}$ .

	$20^{°}$	$40^{°}$	$50^{°}$	$70^{°}$
sin	0·3420	0·6428	0·7660	0·9397
cos	0·9397	0·7660	0·6428	0·3420

41. (a) Using the information given in this triangle, write down, but do not solve, an equation which can be used to find $p$ .

(b) In the given triangle, $A C = 2 cm$ , $B C = 5 cm$ , $\overset{\land}{C} = 90^{°}$ , $\overset{\land}{A} = x^{°}$ and $\overset{\land}{B} = y^{°}$ . Using as much of the information below as is necessary, calculate the value of $x - y$ .

$\sin {23·6}^{°} = 0·4$	$\cos {66·4}^{°} = 0·4$
$\tan {21·8}^{°} = 0·4$

42. The diagram, which is not drawn to scale, represents a crane. The section $A D$ is vertical. The jib, which is hinged at the point $A$ , rotates from the horizontal position $A B$ to the position $A C$ where it is held by a chain $D C$ . Given that $A D = 6 m$ , $A B = A C = 9 m$ and $B \overset{\land}{A} C = 50^{°}$ , calculate

(a) the height of the point $C$ above $A B$ ,

(b) the length of the chain $D C$ ,

(c) the length of the arc $B C$ taking $π$ to be $3·142$ ,

(d) the distance $B C$ .

43. In the diagram, $C \overset{\land}{A} D = 90^{°}$ , $A D = 5 cm$ , $A C = 12 cm$ and $A \overset{\land}{D} B = 45^{°}$ . Calculate

(a) $C D$ ,

(b) the area of $A B C D$ .

44. In the diagram, $B C = 12 cm$ , $A \overset{\land}{B} C = C \overset{\land}{B} D = 60^{°}$ and $A \overset{\land}{C} B = B \overset{\land}{D} C = 90^{°}$ . Using as much of the information given below as is necessary, find

(a) $C D$ ,

(b) $A B$ .

$[\sin 60^{°} = 0·87, \cos 60^{°} = 0·5, \tan 60^{°} = 1·73]$

45. $A B C D E$ is a regular pentagon whose centre is $O$ . The point $M$ is the midpoint of $A B$ .

(a) Show that $A \overset{\land}{O} B = 72^{°}$ .

(b) Given that $O A = O B = 6 cm$ , calculate

(i) $O M$ ,

(ii) $A B$ ,

(iii) the area of the pentagon.

46. (a) $X Y Z$ is a triangle in which $X Y = 8·27 cm$ , $\overset{\land}{X} = 52^{°}$ and $\overset{\land}{Z} = 85^{°}$ . Calculate the length of $Y Z$ .

(b) $P$ , $Q$ and $R$ are three points on level ground. $P Q = 5 km$ , $Q R = 9 km$ and $R P = 10 km$ .

(i) Calculate $\overset{\land}{P}$

(ii) A man walks from $P$ to $Q$ and from $Q$ to $R$ at an average speed of $6 km/h$ . Calculate the total time he takes.

(iii) He then walks directly back from $R$ to $P$ , taking one hour less than on his outward journey from $P$ to $R$ . Calculate the speed at which he returns to $P$ .

(iv) Calculate, in cm, the length of the line representing $Q R$ on a map of scale $1 : 50 000$ .

47. $A B C D$ is a rhombus in which the diagonal $B D = 10 cm$ , the diagonal $A C = 12 cm$ and $A B = x cm$ . Calculate

(a) $x^{2}$ ,

(b) the area of $A B C D$ .

48. In the triangle $A B C$ , $A B = 25 cm$ , $A C = 24 cm$ , $B C = 7 cm$ and $A \overset{\land}{C} B = 90^{°}$ .

(a) Write down the value of $\cos B \overset{\land}{A} C$

(i) as a fraction,

(ii) as a decimal.

(b) Write down the radius of the circle which passes through $A$ , $B$ and $C$ .

49. $X Y Z$ is an isosceles triangle in which $X Y = Y Z = 6 cm$ . $P$ is the foot of the perpendicular from $Y$ onto $X Z$ . Given that $Y \overset{\land}{X} Z = 50^{°}$ , and using as much of the information given below as is necessary, calculate, giving your answers correct to 2 significant figures,

(a) $Y P$ ,

(b) $X Z$ .

$[\sin 40^{°} = 0·643, \cos 40^{°} = 0·766, \tan 40^{°} = 0·839]$

$[\sin 50^{°} = 0·766, \cos 50^{°} = 0·643, \tan 50^{°} = 1·19]$

50. In $Δ A B C$ , $A B = 4 cm$ , $A C = 5 cm$ and $B \overset{\land}{A} C = 100^{°}$ . Using as much of the information given below as is necessary, calculate

(a) the area of $Δ A B C$ ,

(b) $B C^{2}$ .

$[\sin 80^{°} = 0·985, \cos 80^{°} = 0·174]$

51. In the diagram, $G \overset{\land}{H} K = G \overset{\land}{L} K = 90^{°}$ and $K L M$ is a straight line. $G H = 5·5 cm$ , $G K = 7·3 cm$ , $G L = 6·6 cm$ and $G \overset{\land}{M} L = {35·7}^{°}$ .

Calculate

(a) $H K$ ,

(b) $K \overset{\land}{G} L$ ,

(c) $G M$ .

52. In the diagram, $P T R$ is a straight line. $O \overset{\land}{P} T = 18^{°}$ , $P \overset{\land}{T} Q = 150^{°}$ , $Q T = 8 cm$ and $T R = 5 cm$ . Calculate

(a) $P T$ ,

(b) the area of $Δ P Q R$ .

53. Angle $A B C = 90^{°}$ , $A C = 5 cm$ , $B C = 4 cm$ and $B C D$ is a straight line. Calculate

(a) $A B$ ,

(b) $\sin A \overset{\land}{C} D$ ,

(c) $\cos A \overset{\land}{C} D$ .

54. In the triangle $G H K$ , $G H = 2 cm$ , $H K = 5 cm$ and $K G = 4 cm$ . Calculate the value of $\cos \overset{\land}{K}$ , giving your answer as a fraction.

55. (a) In the diagram, $A C = 8 cm$ , $A D = 9·6 cm$ , $A \overset{\land}{B} C = 90^{°}$ , $A \overset{\land}{C} B = 54^{°}$ and $A \overset{\land}{C} D = 74^{°}$ . Calculate

(i) $A B$ ,

(ii) $A \overset{\land}{D} C$ .

(b) In the triangle $P Q R$ , $M$ is the midpoint of $P Q$ . Given that $Q R = 5·3 cm$ , $P \overset{\land}{Q} R = 90^{°}$ and $Q \overset{\land}{R} M = 56^{°}$ , calculate $M \overset{\land}{R} Q$ .

56. In the diagram, $A \overset{\land}{B} C = 90^{°}$ , $B C = 3 cm$ and $D$ is the point on $A B$ such that $D B = 4 cm$ .

(a) Calculate $C D$ ,

(b) Write down, as a fraction, the value of $\tan B \overset{\land}{D} C$ ,

(c) Given also that $A D = \frac{1}{2} B D$ , calculate the area of $Δ A D C$ .

57. $P Q R$ is a triangular region in which $P Q = 5 km$ , $P \overset{\land}{Q} R = 36^{°}$ and $Q R = 6 km$ . Calculate

(a) $P R$ ,

(b) the area of the region, expressing your answer in hectares, correct to the nearest hectare.

$[1 {km}^{2} = 100 hectares]$

58. In the diagram, $A \overset{\land}{B} C = 90^{°}$ , $B C = 8 cm$ and $D$ is the point on $A B$ such that $D B = 6 cm$ . Calculate

(a) $C D$ ,

(b) Write down, as a fraction, the value of $\tan B \overset{\land}{D} C$ ,

(c) Given also that $A D = \frac{1}{2} B D$ , calculate the area of $Δ A D C$ .

59. In the diagram, $A B C$ is a straight line, $D B = 9 cm$ , $B \overset{\land}{C} D = {23·6}^{°}$ and $B \overset{\land}{D} C = {23·6}^{°}$ . Using as much of the information given below as is necessary, calculate

(a) $B C$ ,

(b) $\cos D \overset{\land}{B} C$ ,

(c) $\cos A \overset{\land}{B} D$ .

$[\cos {23·6}^{°} = 0·916, \tan {23·6}^{°} = 0·437]$

60. $A B C$ is a right-angled triangle in which $A \overset{\land}{B} C = 90^{°}$ , $A B = 5 cm$ and $A C = 13 cm$ . The point $D$ lies on $B C$ produced.

(a) Calculate $B C$ .

(b) Write down, as a fraction, the value of

(i) $\tan A \overset{\land}{C} B$ ,

(ii) $\sin A \overset{\land}{C} D$ .

61. Write down a simple geometrical reason why it is not possible to draw

(a) a triangle $A B C$ in which $\overset{\land}{A} = 70^{°}$ , $\overset{\land}{B} = 80^{°}$ and $\overset{\land}{C} = 50^{°}$ ;

(b) a triangle $P Q R$ in which $\overset{\land}{P} = 90^{°}$ , $P Q = 3 cm$ , $Q R = 7 cm$ and $R P = 6 cm$ ;

(c) a triangle $X Y Z$ in which $X Y = 14 cm$ , $X Z = 5 cm$ and $Y Z = 7 cm$ .

62. The diagram shows some of the beams supporting a roof. $A B C$ is a straight line, $A \overset{\land}{B} E = E \overset{\land}{C} D = 90^{°}$ , $C \overset{\land}{D} E = 18^{°}$ , $A E = 9 m$ , $B E = 4 m$ and $E C = 10 m$ . Calculate

(a) $A B$ ,

(b) $B \overset{\land}{C} E$ ,

(c) $C D$ .

63. The triangle $A O B$ lies in a horizontal plane and $T$ is vertically above $O$ . $O A = 8 m$ , $O B = 11 m$ , $O T = 6 m$ , $A T = 10 m$ and $A \overset{\land}{O} B = 90^{°}$ .

(a) Write down the value of $\cos A \overset{\land}{T} O$ , giving your answer as a fraction.

(b) Calculate the volume of the pyramid $O A B T$ .

$[Volume of a pyramid = \frac{1}{3} area of base \times perpendicular height]$

64. In the diagram, $O$ is the centre of the circle through $A$ , $B$ and $C$ , angle $O C B = 50^{°}$ and $B C X$ and $A O C$ are straight lines.

Using as much of the information below as is necessary, find

(a) Calculate $A \overset{\land}{B} O$ ,

(b) Using as much of the information as possible, find:

(i) $\sin A \overset{\land}{C} X$ ,

(ii) $\cos A \overset{\land}{C} X$ .

$[\sin 40^{°} = 0·6428, \cos 40^{°} = 0·7660]$

$[\sin 50^{°} = 0·7660, \cos 50^{°} = 0·6428]$

65. The rectangular side $A B C D$ of a large lorry is strengthened by three metal struts $P Q$ , $Q D$ and $D P$ . Given that $A P = P B = 2 m$ , $B Q = 3 m$ and $P D = 8 m$ , calculate

(a) $P Q$ ,

(b) $P \overset{\land}{D} A$ ,

(c) the area of $Δ A P D$ ,

(d) $C \overset{\land}{Q} D$ .

66. In the diagram $D \overset{\land}{A} B = D \overset{\land}{B} C = 90^{°}$ , $A B = 16 cm$ , $B D = 20 cm$ , $A D = 12 cm$ and $A B$ is parallel to $D C$ .

(a) Express $\tan A \overset{\land}{B} D$ as a fraction in its lowest terms,

(b) Calculate $B C$ ,

(c) Calculate the area of trapezium $A B C D$ .

67. In the diagram, $C \overset{\land}{B} D = 30^{°}$ , $C \overset{\land}{D} B = 80^{°}$ , $C D = 10 cm$ and $A B D E$ is a straight line. Using as much of the information from the table given below as is necessary

(a) find

(i) $\sin A \overset{\land}{B} C$ ,

(ii) $\cos C \overset{\land}{D} E$ ,

(b) calculate $B C$ .

	$30^{°}$	$70^{°}$	$80^{°}$
sin	0·500	0·940	0·985
cos	0·866	0·342	0·174
tan	0·577	2·747	5·671

68. In the diagram, $A C = 8·4 cm$ , $A D = 12 cm$ , $A \overset{\land}{C} B = A \overset{\land}{C} D = 90^{°}$ and $A \overset{\land}{B} C = 30^{°}$ . Calculate

(a) $A B$ ,

(b) $D \overset{\land}{A} C$ ,

(c) the area of triangle $C A X$ , where $X$ is the midpoint of $A D$ .

69. In the right angled triangle $A B C$ , $P$ is a point on the side $A B$ . Given that $A P = 4 cm$ , $P B = 5 cm$ , $B C = 12 cm$ and $P C = 13 cm$ , calculate

(a) $A C$ ,

(b) $\cos B \overset{\land}{P} C$ ,

(c) $\tan P \overset{\land}{A} C$ ,

(d) $\sin A \overset{\land}{P} C$ .

70. A ladder $A B$ is $7 m$ long. Each of the diagrams below shows the ladder in a different position, in each case the foot of the ladder is on horizontal ground.

(a) In diagram I, the ladder leans against a vertical wall and makes an angle of $68^{°}$ with the ground. Calculate $B W$ , the height of the top of the ladder above the ground.

(b) In diagram II, the ladder rests against the roof of a shed $5 m$ high and projects $1·5 m$ above it. Calculate $B \overset{\land}{A} X$ , the angle that the ladder makes with the ground.

(c) In diagram III, the ladder is supported away from a vertical wall by a horizontal rod $C D$ , $0·9 m$ long. The point $C$ is $0·9 m$ from the top of the ladder and the ladder makes an angle of $75^{°}$ with the ground. Calculate $A Z$ , the distance from the foot of the ladder to the wall.

71. In the diagram, $A D = 2·8 cm$ , $A X = 5·5 cm$ and $B X = 4·1 cm$ .
$A \overset{\land}{D} C = 90^{°}$ , $D \overset{\land}{A} X = 74^{°}$ , $A \overset{\land}{X} B = 123^{°}$ $A X C$ is a straight line.

(a) Calculate

(i) $A C$ ,

(ii) $A B$ .

(b) Given that $Y$ is the point on $A X$ such that $D Y$ is parallel to $X B$ , calculate $A Y$ .

72. In triangle $A B C$ , angle $B = 70^{°}$ and angle $C = 80^{°}$ . The point $N$ lies on $A B$ such that angle $A N C = 90^{°}$ .

(a) Calculate angle $B C N$ .

(b) Given also that $X$ lies on $A C$ such that $N X$ is parallel to $B C$ , calculate

(i) angle $A X N$ ,

(ii) angle $B N X$ .

73. In the diagram, $P Q = 24 cm$ , $Q R = 7 cm$ , $R S = 23 cm$ , $P R = 25 cm$ , $P \overset{\land}{Q} R = 90^{°}$ and $Q R S$ is a straight line. Giving each answer as a fraction, find

(a) $\sin Q \overset{\land}{P} R$ ,

(b) $\tan Q \overset{\land}{S} P$ ,

(c) $\cos P \overset{\land}{R} S$ .

74. In the trapezium $A B C D$ , $A B$ is parallel to $D C$ , $A \overset{\land}{D} C = 90^{°}$ , $C \overset{\land}{A} B = 32^{°}$ , $A \overset{\land}{C} B = 112^{°}$ and $A C = 5·38 cm$ . Calculate

(a) $C D$ ,

(b) $C B$ .

75. The quadrilateral $A B C D$ represents a kite which is symmetrical about the line $B D$ . Given that $A D = 60 cm$ , $B D = 70 cm$ and $A \overset{\land}{D} B = 23^{°}$ , calculate

(a) $A B$ ,

(b) $A C$ .

76. In the diagram, $B \overset{\land}{A} D = A \overset{\land}{C} D = 90^{°}$ , $A \overset{\land}{D} B = 23^{°}$ , $A D = 6 cm$ and $B C D E$ is a straight line. Using as much of the information given below as is necessary, calculate

(a) $A C$ ,

(b) $C D$ ,

(c) $A B$ ,

(d) $\sin A \overset{\land}{D} E$ .

$[\sin 23^{°} = 0·391, \cos 23^{°} = 0·921, \tan 23^{°} = 0·424]$

77. $A B$ , $B C$ , $C D$ , $A D$ , $A C$ and $C E$ are some of the beams supporting a roof. $A D$ and $B C$ are horizontal and $A C$ is vertical. $C E$ is perpendicular to $A B$ . $A C = 3·9 m$ , $C D = 4·4 m$ and $B \overset{\land}{A} C = 58^{°}$ . Calculate

(a) the length of $C E$ ,

(b) the length of $A B$ ,

(c) $A \overset{\land}{C} D$ .

78. In the diagram, $B C D$ is a straight line. $A B = 90 m$ , $A C = 80 m$ , $C D = 120 m$ and $B \overset{\land}{C} A = 116^{°}$ .

(a) Calculate

(i) $B \overset{\land}{A} C$ ,

(ii) $A D$ .

(b) The point $E$ , inside triangle $A C D$ , is such that $C E = 62 m$ and the area of triangle $C D E$ is $2200 m^{2}$ . Calculate $E \overset{\land}{C} D$ .

79. In the diagram, $P \overset{\land}{Q} R = Q \overset{\land}{S} R = 90^{°}$ , $Q \overset{\land}{P} S = {21·1}^{°}$ and $P Q = 7 cm$ . $P S R$ and $Q P X$ are straight lines. Using as much of the information given below as is necessary, calculate

(a) $Q R$ ,

(b) $Q S$ ,

(c) $\cos P \overset{\land}{Q} S$ ,

(d) $\sin R \overset{\land}{P} X$ .

$[\sin {21·1}^{°} = 0·360, \cos {21·1}^{°} = 0·933, \tan {21·1}^{°} = 0·386]$

80. In the diagram $Q R S$ is a straight line, $P Q = 12 m$ , $Q R = 5 m$ and $R P = 13 m$ .

(a) Explain why $P \overset{\land}{Q} R$ is a right angle.

(b) Expressing your answer as a fraction, write down

(i) $\tan Q \overset{\land}{P} R$ ,

(ii) $\cos P \overset{\land}{R} S$ .

81. The diagram represents a framework. Given that $B \overset{\land}{A} C = 52^{°}$ , $A \overset{\land}{B} C = C \overset{\land}{D} E = A \overset{\land}{C} E = 90^{°}$ , $A C = 3 m$ and $E C = 1·2 m$ , calculate

(a) $B C$ ,

(b) $E \overset{\land}{A} C$ ,

(c) $C D$ .

82. A thin string of length $50 cm$ has one end fastened to a point $A$ , on the rim of a circular wheel of radius $20 cm$ and centre $C$ . The other end is fastened to a small bead, $B$ , which is threaded on a thin fixed wire $P Q$ . $Q$ touches the wheel and $P$ , $Q$ and $C$ lie on a straight line. The wheel is slowly rotated clockwise, thus pulling the bead along the wire.

(a) When $A$ is at the position $A_{1}$ , the bead is at the position $B_{1}$ and $A_{1} \overset{\land}{C} B_{1} = 36^{°}$ . Find $A_{1} \overset{\land}{B_{1}} C$ , the angle between the string and the wire.

(b) When $A$ is at the position $A_{2}$ , the bead is at the position $B_{2}$ and $B_{2} \overset{\land}{A_{2}} C = 125^{°}$ . Find $B_{2} C$ , the distance of the bead from the centre of the wheel.

(c) When $A$ is at the position $A_{3}$ , the bead is at the position $B_{3}$ and part of the string, $B_{3} T$ , is tangent to the circle. The length of the string around the arc $T A_{3}$ is $15 cm$ . Taking the value of $π$ to be $3·142$ , calculate

(i) $A_{3} \overset{\land}{C} T$ ,

(ii) $B_{3} Q$ , the distance of the bead from the wheel.

83. $A B C$ is a triangle in which $B \overset{\land}{A} C = 90^{°}$ , $A C = 40 cm$ and $B C = 41 cm$ . $A B$ is produced to $P$ and $A C$ is produced to $Q$ .

(a) Showing your working clearly, explain why $A B = 9 cm$ .

(b) Express as a fraction

(i) $\tan B \overset{\land}{C} A$ ,

(ii) $\sin P \overset{\land}{B} C$ ,

(iii) $\cos B \overset{\land}{C} Q$ .

84. $A C$ is a diameter of a circle and $B$ and $D$ are points on the circumference. The tangent to the circle at the point $C$ meets $A D$ produced at $T$ .

(a) State briefly a reason why $A \overset{\land}{B} C = 90^{°}$ .

(b) Given that $A B = 7 cm$ , $A C = 12 cm$ and $C \overset{\land}{A} D = 19^{°}$ , calculate

(i) $B C$ ,

(ii) $A \overset{\land}{C} B$ ,

(iii) $C D$ ,

(iv) $A T$ .

85. $E F G$ is a triangle in which $E \overset{\land}{F} G = 90^{°}$ , $E F = 24 cm$ and $E G = 25 cm$ . $F E D$ and $F G H$ are straight lines.

(a) Showing your working clearly, explain why $G F = 7 cm$ .

(b) Express as a fraction

(i) $\tan E \overset{\land}{G} F$ ,

(ii) $\sin E \overset{\land}{G} H$ ,

(iii) $\cos D \overset{\land}{E} G$ .

86. $A B C$ is a triangle in which $A \overset{\land}{C} B = 90^{°}$ . $D$ is a point on $A C$ . $A B = 20 cm$ , $B C = 12 cm$ , $C D = 5 cm$ , $B D = 13 cm$ and $D A = 11 cm$ . Giving each answer as a fraction, find

(a) $\tan C \overset{\land}{D} B$ ,

(b) $\cos C \overset{\land}{A} B$ ,

(c) $\sin A \overset{\land}{D} B$ .

87. In the diagram, $B C D$ is a straight line, $B C = 6 cm$ , $A C = 7·9 cm$ , $A D = 6·4 cm$ and $A \overset{\land}{C} B = 130^{°}$ . Calculate

(a) $A B$ ,

(b) the area of triangle $A B C$ ,

(c) the shortest distance from $B$ to $A C$ produced,

(d) the obtuse angle $A \overset{\land}{D} C$ .

88. In the diagram, $B C D$ is a straight line, $B C = 4 cm$ , $A D = 10 cm$ , $A C = 6·4 cm$ , $A \overset{\land}{C} D = A \overset{\land}{E} D = 90^{°}$ and $D \overset{\land}{A} E = 47^{°}$ . Calculate

(a) $A E$ ,

(b) $A \overset{\land}{D} C$ ,

(c) $A \overset{\land}{B} C$ .

89. $A B C D$ represents a rectangular gate. $P$ lies on $B C$ and $Q$ lies on $C D$ . The lines $A P$ , $A C$ and $A Q$ represent metal struts. $A B = 2 m$ , $A D = 1·1 m$ , $B \overset{\land}{A} P = 15^{°}$ and $A \overset{\land}{Q} D = 57^{°}$ .

Calculate

(a) $B P$ ,

(b) $A C$ ,

(c) $A Q$ ,

(d) $P \overset{\land}{A} C$ .

90. In the diagram, $B D = 8 cm$ , $B C = 10 cm$ , $A \overset{\land}{B} D = 90^{°}$ , $A \overset{\land}{D} B = 40^{°}$ and $D \overset{\land}{B} C = 130^{°}$ . Using as much of the information below as is necessary, calculate

(a) $A B$ ,

(b) the area of triangle $B C D$ .

	$\sin$	$\cos$	$\tan$
$40^{°}$	0·643	0·766	0·839
$50^{°}$	0·766	0·643	1·192

91. $A$ , $B$ and $C$ lie in a straight line on level ground. $T$ is the top of a vertical flagpole $T C$ .

(a) John wants to find the height of the flagpole. He measures the angle of elevation of the top of the flagpole from $A$ and finds that it is $32^{°}$ . He then walks $7 m$ to $B$ and finds that the angle of elevation is now $40^{°}$ . Calculate

(i) $A \overset{\land}{T} B$ ,

(ii) the length $B T$ ,

(iii) the height, $T C$ , of the flagpole.

(b) On the opposite side of the flagpole from $A$ and $B$ , the ground slopes down to $D$ such that $T \overset{\land}{C} D = 106^{°}$ . A rope is stretched from $D$ , which is $15 m$ from $C$ , to the point $R$ , where $C R = 12 m$ . Calculate the length of the rope $D R$ .

(c) A second vertical flagpole $D E$ is to be erected at $D$ . Given that $R E$ is horizontal, calculate the length $D E$ .

92. The diagram shows a frame that is used to support a hanging weight. $A B C$ is a vertical beam and $B E D$ is a horizontal beam. $A D,$ $A E$ and $C D$ are three supporting struts. Given that $B E = 2 m$ , $B C = 5 m$ , $A E = 6 m$ and angle $B \overset{\land}{D} C = 35^{°}$ , calculate

(a) $A B$ ,

(b) $C D$ ,

(c) $E D$ ,

(d) $B \overset{\land}{D} A$ .

93.

In each of the diagrams above, the point $R$ represents a rock on the bottom of a lake, the point $B$ represents a boat on the surface, directly above the rock, and the point $D$ represents the position of a diver. Both the diver and the rock are $22 m$ below the surface and the bottom of the lake is horizontal. The diver is connected to the boat by a thin rope. This rope is kept straight and is represented by the line $D B$ .

Each of the Diagrams I, II and III represents a different situation.

(a) (i) In Diagram I, the rope makes an angle of $38^{°}$ with the vertical. Calculate $D R$ , the distance from the diver to the rock.

(ii) In Diagram II, the length of the rope is $50 m$ . Calculate $D \overset{\land}{B} R$ , the angle the rope makes with the vertical.

(iii) In Diagram III, the rope makes an angle of $70^{°}$ with the horizontal. Calculate $D B$ , the length of the rope.

(b) Describe the set of points (in three dimensions), which the diver $D$ can reach, given that the boat $B$ is fixed and that the rope $D B$ is $15 m$ long and is kept straight.

94. In the diagram, $A$ , $B$ and $C$ represent three towns. They are joined by straight roads. The distance $A C = 20 km$ , $B C = 15 km$ and $A \overset{\land}{C} B = 110^{°}$ .

(a) Calculate

(i) the area of triangle $A B C$ ,

(ii) the distance $A B$ ,

(iii) the shortest distance from $C$ to the roads $A B$ .

(b) A picnic place is situated at $P$ where $P \overset{\land}{C} B = 18^{°}$ and $B \overset{\land}{P} C = 140^{°}$ . Calculate the distance $B P$ .

95. The diagrams shows the paths in a park. $A B C$ is a straight line. Angle $E A D = angle A B D = 90^{°}$ , angle $A D = 37^{°}$ and angle $B \overset{\land}{D} C = 56^{°}$ . $B D = 420 m$ and $A D = 550 m$ . Calculate

(a) $A B$ ,

(b) $B C$ ,

(c) $D E$ .

96. In the diagram, $A$ , $B$ , $C$ and $D$ are four markers on a horizontal field. $B D = 15 m$ , $D C = 12 m$ , $A \overset{\land}{B} D = 35^{°}$ , $A \overset{\land}{D} B = 90^{°}$ and $B \overset{\land}{D} C = 160^{°}$ .

(a) Calculate the distance $A B$ .

(b) Calculate the distance $B C$ .

97. In the diagram $A C X$ is a straight line. $A B = 12 cm$ , $\sin B \overset{\land}{A} C = \frac{1}{2}$ and $\sin B \overset{\land}{C} A = \frac{3}{4}$ .

(a) Write down the value of $\sin B \overset{\land}{C} X$ .

(b) Calculate $B C$ .

98. A crane stands on level ground. It may be represented by a tower $A B C D$ , of height $11 m$ , and a jib $B R$ . The jib is of length $20 m$ and can rotate in a vertical plane about $B$ . A vertical cable, $R S$ , carries a load $S$ . The diagrams show two possible positions of the jib, cable and load.

(a) Diagram I shows the situation when $B S$ is horizontal and $R S = 8 m$ . Calculate

(i) the distance $B S$ ,

(ii) the angle that the jib, $B R$ , makes with the horizontal.

(b) Diagram II shows another situation. The jib, $B R$ , has been rotated and the length $R S$ increased. The load, $S$ , is now on the ground at a point $5 m$ from $A$ . Calculate

(i) the angle through which the jib has been rotated,

(ii) the length by which $R S$ has increased.

99.

In the diagram $A B C$ is a straight line. $D$ is a point such that $C D = 6 cm$ , $\sin B \overset{\land}{C} D = 0·5$ and $\sin D \overset{\land}{B} C = 0·8$ .

(a) Calculate $B D$ .

(b) Write down the value of $\sin A \overset{\land}{B} D$ .

100. $A B C$ is a triangle with $A B = 5 cm$ , $B C = 4 cm$ and angle $A \overset{\land}{B} C = 120^{°}$ . $A B$ is produced to $D$ and angle $B \overset{\land}{C} D = 90^{°}$ . Using as much information given in the table below as is necessary,

	$\sin$	$\cos$	$\tan$
$120^{°}$	0·87	-0·5	-1·73

calculate

(a) the area of triangle $A B C$ ,

(b) the length of $B D$ .

101. Three buoys, $A$ , $B$ and $C$ , are positioned in a lake to provide a course for a yacht race.

$A B = 800 m$ , $A \overset{\land}{B} C = 32^{°}$ , $B \overset{\land}{A} C = 22^{°}$ and $N$ is the point on $A B$ which is $200 m$ from $A$ .

(a) Show that the distance $A C$ is $524 m$ , correct to the nearest metre.

(b) Calculate the distance $N C$ .

102. In the diagram $A C = 26 cm$ , $A \overset{\land}{D} C = 90^{°}$ , $D \overset{\land}{C} A = A \overset{\land}{C} B$ and $C B X$ is a straight line. Using as much information given in the table below as is necessary,

(a) calculate the length of $D C$ ,

(b) calculate the length of $C B$ ,

(c) write down the value of $\cos A \overset{\land}{B} X$ .

	$\sin$	$\cos$	$\tan$
$a^{°}$	$\frac{56}{65}$	$\frac{33}{65}$	$\frac{56}{33}$
$b^{°}$	$\frac{4}{5}$	$\frac{3}{5}$	$\frac{4}{3}$
$c^{°}$	$\frac{12}{13}$	$\frac{5}{13}$	$\frac{12}{5}$

103. A triangular sheet of cardboard, $A B C$ , has sides whose lengths are $A B = 31 cm$ , $A C = 53 cm$ and $B C = 47 cm$ .

(a) Calculate $A \overset{\land}{C} B$ , giving your answer correct to one decimal place.

(b) A triangle $A B D$ is to be cut from triangle $A B C$ . The area of triangle $A B D$ is $300 {cm}^{2}$ and the length of $B D$ is $23 cm$ .

Calculate $A \overset{\land}{B} D$ , giving your answer correct to one decimal place.

104. In the diagram, $B C D E$ is a straight line. $B C = 7 cm$ and $C D = 5 cm$ . $A$ is the point such that $A \overset{\land}{B} C = 90^{°}$ , $A B = 5 cm$ and $A D = 13 cm$ .

(a) Write down the value of

(i) $\cos A \overset{\land}{D} B$ ,

(ii) $\tan C \overset{\land}{A} B$ ,

(iii) $\sin A \overset{\land}{D} E$ .

(b) Calculate the numerical value of $A C^{2}$ .

105. In the diagram, $A B D$ is a straight line, $A B = 10 cm$ , $B C = 6 cm$ and $B \overset{\land}{C} D$ is a right angle. $C \overset{\land}{B} D = x^{°}$ , where $\sin x^{°} = 0.6$ , $\cos x^{°} = 0.8$ and $\tan x^{°} = 0.75$ . Calculate

(a) $C D$ ,

(b) $\cos A \overset{\land}{B} C$ ,

(c) the area of triangle $A B C$ .

106. In the diagram, $B C D .$ is a straight line, $B A$ is parallel to $C E$ , $E D = C D$ , $B \overset{\land}{A} C = 40^{°}$ , $A \overset{\land}{B} C = 72^{°}$ , $C \overset{\land}{E} A = 82^{°}$ .

(a) Calculate

(i) $A \overset{\land}{C} E$ ,

(ii) $C \overset{\land}{A} E$ ,

(iii) $C \overset{\land}{D} E$ .

(b) Given that $B C = 8 cm$ , calculate the length of $A C$ .

107. In the triangle shown, $X Y = 5 cm$ , $Y Z = 7 cm$ and $Z X = 6 cm$ . Calculate $X \overset{\land}{Y} Z$ .

108. The diagram shows three points $A (- 2, - 1)$ , $B (1, - 1)$ and $C (4, 3)$ . Calculate

(a) the area of triangle $A B C$ ,

(b) the cosine of $A \overset{\land}{B} C$ .

109. A ladder $F T$ stands on horizontal ground at $F$ and leans against a vertical wall at $T$ . The point $W$ , on the ground, is vertically below $T$ . The ladder can be extended to various lengths. The diagrams above show three positions of this ladder.

(a) In Diagram I, $F T = 5.5 m$ and $T \overset{\land}{F} W = 65^{°}$ . Calculate $F W$ .

(b) In Diagram II, $T W = 7.3 m$ and $F W = 2.6 m$ . Calculate the angle which the ladder makes with the ground.

(c) In Diagram III, $T W = 9.3 m$ and $T \overset{\land}{F} W = 71^{°}$ . Calculate by how much the ladder has been extended from its original length of $5.5 m$ .

110. The vertices of triangle $L M N$ are $(- 1, - 2)$ , $(2, 5)$ and $(2, 2)$ respectively. $K$ is the point $(2, - 2)$ . Calculate the value of

(a) $\tan L \overset{\land}{N} K$ ,

(b) $\cos L \overset{\land}{N} M$ .

111. (a) $A B C$ is an equilateral triangle of side $2$ units. $A N$ is perpendicular to $B C$ . Show that $\cos 60^{°} = \frac{1}{2}$ .

(b) In the diagram, which is not accurate:

$R R^{'}$ is a part of the locus of points equidistant from $A$ and $B$ ,

$S S^{'}$ is a part of the locus of points equidistant from the lines $A B$ and $B C$ and the point $M$ lies on both loci.

It is given that $A B = 20 cm$ and $A \overset{\land}{B} C = 120^{°}$ .

Calculate $B M$ , showing your working clearly.

112. In the diagram, $A B C$ is a straight line, $C \overset{\land}{B} D = {68.2}^{°}$ , $B \overset{\land}{C} D = 90^{°}$ , $C D = 5 cm$ and $A D = 8 cm$ . Using as much of the information given in the table below as is necessary, find

	$\sin$	$\cos$	$\tan$
${68.2}^{°}$	$0.93$	$0.37$	$2.50$

(a) $\sin x$ , giving your answer as a fraction,

(b) $\cos y$ ,

(c) $B C$ .

113. The diagram shows footpaths $B R$ and $C R$ in a park $A B C D R$ . $B R = 140 m$ , $C R = 120 m$ and $A B = 82 m$ . $B \overset{\land}{C} R = 48^{°}$ , $D \overset{\land}{C} R = 59^{°}$ and $A \overset{\land}{B} R = C \overset{\land}{D} R = 90^{°}$ .

Calculate

(a) $C D$ ,

(b) $A \overset{\land}{R} B$ ,

(c) $C \overset{\land}{B} R$ .

114. The diagram shows three points $A (- 2, 7)$ , $B (- 2, 2)$ and $C (6, - 4)$ .

Find

(a) the length $B C$ ,

(b) the area of triangle $A B C$ ,

(c) the value of $\sin A \overset{\land}{B} C$ .

115. In the diagram, $A B$ is a vertical wall. A beam, $C D$ , of length $11 metres$ , rests with one end, $D$ , on horizontal ground. It is held in place by two cables, $B C$ and $B D$ . Given that $A D = 8 metres$ , $B D = 15 metres$ and angle $B \overset{\land}{D} C = 55^{°}$ , calculate

(a) the length of $A B$ ,

(b) the length of the cable $B C$ ,

(c) the angle between the beam $C D$ and the ground.

116. The diagram shows the points $P (3, 7)$ , $Q (7, 7)$ and $R (- 2, - 5)$ .

Find

(a) the length $P R$ ,

(b) the area of triangle $P Q R$ ,

(c) the value of $\sin Q \overset{\land}{P} R$ .