# GEOMETRIC MENSURATION

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1 GEOMETRI MENSURTION

2 Question 1 (**) 8 cm 6 cm θ 6 cm O The figure above shows a circular sector O, subtending an angle of θ radians at its centre O. The radius of the sector is 6 cm and the length of the chord is 8 cm. a) Find the size of the angle θ in radians, correct to two decimal places. b) Determine the area of the circular segment, shown shaded in the figure., c θ 1.46, area 8.38 to 8.39

3 Question (**) O 5cm 1.8 5cm The figure above shows a circle with centre at O and radius 5 cm. The points and lie on the circle so that the angle O is 1.8 radians. a) Find the area of the sector O. b) Determine the length of the chord. c) Hence show that the perimeter of the minor segment, shown shaded in the figure, is approximately 16.8 cm. area =.5, 7.83

4 Question 3 (**) 1.8 O D 0cm 5cm The figure above shows two concentric circular sectors O and OD, where O is their common centre. oth sectors subtend an angle of 1.8 radians at O. The point lies on O and similarly the point lies on OD. It is further given that O = O = 0cm and O = OD = 5cm. The finite region D is shown shaded in the above figure. Determine the perimeter and the area of D. P = 91 cm, = 0.5 cm

5 Question 4 (**) 7cm 8cm 5cm The figure above shows a triangle where the following information is given. = 7 cm, = 8 cm and = 5 cm. Find the size of the angle in degrees, and hence determine as an exact surd the area of the triangle. = 60, rea = 10 3 cm

6 Question 5 (**) 70 m 37 m θ O 37 m The figure above shows a circular sector O, subtending an angle of θ radians at its centre O. The radius of the sector is 37 m and the length of the chord is 70 m. a) Show that θ is approximately.481 radians. b) alculate to an appropriate degree of accuracy i. the length of the arc. ii. the shortest distance from O to the chord. iii. the area of the circular segment, shown shaded in the figure. E, 91.8m, 1 m, area 178 m

7 Question 6 (**) r cm r cm c θ The figure above shows a circular sector of radius r cm subtending an angle θ radians at. The length of the arc is 9 of the perimeter of the sector. Show that θ = 4 radians. 7 proof

8 Question 7 (**) 7cm c θ 7cm The figure above shows a circular sector of radius 7 cm subtending an angle θ radians at. Given the perimeter of the sector is numerically equal to the area of the sector show that θ is 0.8 radians. proof

9 Question 8 (**) c.5 The figure above shows a circular sector subtending an angle of.5 radians at the point. Given that the area of the sector is 45 cm, find its perimeter. P = 7cm

10 Question 9 (**) 1m 150 The triangle is such so that is 1 m and the angle is 150. a) Given that the area of the triangle is 30 m, show that the length of is 10 m. b) Find the length of, giving the answer in m, correct to decimal places. c) alculate the smallest angle of the triangle, giving the answer in degrees, correct to one decimal place. 1.6m, 13.6

11 Question 10 (**) 10 cm 15 cm D 9 cm The figure above shows a right angled trapezium D. It is given that = 15 cm, D = 9 cm, D = 10 cm and = D = 90. Determine the length of the straight line. = 17cm

12 Question 11 (**+) R D 1cm The figure above shows a right angled triangle, where the angle is 30. The point D lies on so that the angle D is 75. The length of D is 1 cm. alculate the length of. =

13 Question 1 (**+) 6cm 10cm 45 D The figure above shows the right angled triangle where is 10 cm, is 6 cm and the angle is 90. The point D lies on so that the angle D is 45. Find the area of the triangle D, shown shaded in the figure above. 70

14 Question 13 (**+) 6 3 cm O The figure above shows a badge in the shape of a circular sector O, centred at O. The triangle O is equilateral and its perpendicular height is 6 3 cm. a) Find the length of O. b) Determine in terms of π i. the area of the badge. ii. the perimeter of the badge. O = 1, area = 4π, perimeter = 4 + 4π

15 Question 14 (**+) 4 D 13 1 The figure above shows the triangle, where = 4, = 1 and = 13. a) Show that = π. 3 circular sector D, where D lies on, is drawn inside the triangle. The centre of the sector is at and its radius is 1. b) Determine the area of the shaded region D., area 1.1

16 Question 15 (**+) D 6cm R E 8cm c 1. The figure above shows a triangle where the lengths of and are 8 cm and 6 cm, respectively. The angle is 1. radians. a) Find the length of. b) Determine the area of the triangle. circular arc with centre at and radius 4 cm is drawn inside the triangle. The arc intersects the triangle at the points D and E. The shaded region R is bounded by the straight lines E,, D and the arc ED. c) alculate the area of R. d) alculate the perimeter of R., 8.08, area.4, area 1.8, perimeter 18.9 R R

17 Question 16 (**+) D 5cm θ F 10cm θ E 5cm The figure above shows a rectangle FE with two identical circular sectors attached to its sides F and E. Each of these circular sectors has radius 5 cm and subtends an angle of θ radians at its respective centre, F and E. The length of FE is 10 cm. Given that the perimeter of the entire shape DEF is 37. cm, show clearly that its area is 68 cm. proof

18 Question 17 (**+) x cm 150 ( x + ) cm The figure above shows a triangle whose area is 0 The lengths of and are x cm and ( x + ) angle is 150. a) Find the value of x. b) Determine the length of. cm. cm respectively, and the size of the x = 8, 17.4

19 Question 18 (***) 6cm O c cm R The figure above shows a triangle O where O = 6 cm, O = 10 cm. The angle O is 0.8 radians. a) alculate the area of the triangle O. n arc centred at O with radius 6 cm is drawn inside the triangle, meeting O at. The shaded region R is bounded by, O and the arc. b) Find the area of R. c) Determine the perimeter of R. R, area of triangle 1.5 cm, area of R 7.1 cm, perimeter of R 16.0 cm

20 Question 19 (***) 1cm 1cm D E 1cm 1cm 1cm O 1cm The figure above shows an equilateral triangle of side length cm. The points O, D and E are the midpoints of, and, respectively. circular arc, centred at O, having OD and OE as radii is drawn. Determine the exact area of the shaded region. ( π )

21 Question 0 (***) r cm θ r cm O The figure above shows a circle with centre at O and radius r cm. The minor sector O subtends an angle of θ radians at O. The area of the minor sector O is 48cm. The length of the minor arc is 1cm. Determine the value of r and the value of θ. r = 8, θ = 1.5

22 Question 1 (***) y ( 9,40) ( 7,40) O D( 18,0) ( 36,0) x The figure above shows the cross section of a river dam modelled in a system of coordinate axes where all units are in metres. The cross section of the dam consists of a circular sector D and two isosceles triangles OD and D. The coordinates of the points,, and D are ( 9,40 ), ( 7,40 ), ( ) ( 18,0 ), respectively. a) Find the length of D. b) Show that the angle D is approximately radians. 36,0 and c) Hence determine to the nearest m the cross sectional area of the dam. D = 41, area 109

23 Question (***) 6cm 6cm R c cm 6cm D The figure above shows a rhombus D with side length 6 cm. The angle D is 0.75 radians. circular arc D is drawn inside the rhombus with centre at and radius 6 cm. The arc D divides the rhombus into two regions, the smaller of the two regions shown shaded in the figure, is denoted by R. Find, to three significant figures, the area of R. area 11.0 cm

24 Question 3 (***) D 10 cm 6 cm c θ O The figure above shows two circular arcs and D, which are parts of circular sectors whose centre is at O. oth sectors subtend an angle θ radians at O. OD is a straight line segment with O = 6cm and O = 10cm. Given that the area of the shaded region D is 4 cm, calculate the perimeter of D. perimeter = 0 cm

25 Question 4 (***) 4 cm 48 cm O θ 4 cm 60 cm D The figure above shows a circular sector O whose centre is at O. The radius of the sector is 60 cm. The points and D lie on O and O respectively, so that O = OD = 4 cm. Given that the length of the arc is 48 cm, find the area of the shaded region D, correct to the nearest cm. Q, area = 133

26 Question 5 (***) 5 6 π.4 cm O.4cm The figure above shows a composite shape. The composite shape consists of a circular sector O centred at O, where it subtends an angle of 5 π radians. 6 The straight sides of the sector have length of.4 cm. The triangle O is right angled at and is attached to the sector so that O is a straight line. Find, to two decimal places, the area of the composite shape. V, area 8.79 cm

27 Question 6 (***) 3 π 3 3 The figure above shows the cross section of a railway tunnel, modelled as the major segment of a circle, centre at and radius of 3 m. The angle is π radians. 3 a) Find the exact length of. b) Determine the area of the triangle. c) Show that the cross sectional area of the tunnel is 6π D, = 3 3, area = 9 3 4

28 Question 7 (***+) 3x x 1 O θ x + The figure above shows a circular sector O of radius 3x cm, subtending an angle θ radians at O. The line is perpendicular to O and has length ( x 1) cm. The length of O is ( x + ) cm. a) Show that x = 5. b) Find the area of the shaded region. area 18.4

29 Question 8 (***+) D R c cm The figure above shows a quarter circle D of radius 10 cm, whose centre is at. The point lies on the arc D so that the angle is 0.3 radians. The segment bounded by the semicircle and the chord D is denoted by R. a) Determine the area of R. b) Find the perimeter of R. area 15.8 cm, perimeter 4.6 cm

30 Question 9 (***+). m O 1.8 m D The figure above shows a design of a door. The door design consists of two congruent right angled triangles O and DO where O = DO = 90, and a circular sector O centred at O, where O is the midpoint of D. a) Show that the angle O is approximately radians. b) Hence determine i. the perimeter of the door design. ii. the area of the door design. I, P 8.05 m, 4.17 m

31 Question 30 (***+) r m O θ r m D circular sector OD, subtending an angle θ radians at its centre O, has a radius of r m. The sector has an area of 0.5 Determine the values of r and θ. m and a perimeter of m. c r = 0.5 m, θ =

32 Question 31 (***+) D O c 0.6 1cm The figure above shows a semi circle with centre at O and radius 1 cm. The diameter of the semicircle is O, the chord D is parallel to O and the angle c DO is 0.6. a) Find the area of the shaded segment. b) Find the perimeter of the shaded segment. area 7.7 cm, perimeter 43.1 cm

33 Question 3 (***+) ( x + 4) cm ( x + 3) cm 60 ( x +1) cm The figure above shows a triangle whose side lengths are given in terms of x. Given that the angle is 60, determine a) the value of x. b) the exact area of the triangle. x = 4, area = 10 3

34 Question 33 (***+) ( 3x + 1) cm ( x + 9) cm 60 ( x) cm The figure above shows a triangle whose side lengths are given in terms of x. Given that the angle is 60, determine the exact area of the triangle. area = 40 3

35 Question 34 (***+) In the triangle = cm, = 4 cm and = 3 cm. Find the exact area of the triangle. H, area =

36 Question 35 (***+) O 1cm 1 3 cm The figure above shows a circle with centre at O and radius 1 cm. The chord has a length of 1 3 cm. a) Show that the angle O is π radians. 3 b) Find, in exact form, the area of the major segment bounded by the chord. area = 1( π )

37 Question 36 (***+) 5cm 6cm O c 1. 5cm The figure above shows a template design. The curve is the arc of a circular sector O, subtending an angle of 1. radians at its centre O. The radius of the sector is 5 cm. The straight lines and are of equal length. The length of the straight line O is 6 cm. Find, to three significant figures where appropriate, a) the area of the circular sector O. b) the size of the angle O, in radians. c) the total area of the template design. area = 15 cm, c O.54, area cm =

38 Question 37 (***+) 7 cm θ x cm 3x cm 60 The figure above shows a triangle with side lengths = 7 cm, = x cm and = 3x cm. The sizes of the angles and are 60 and θ, respectively. y using the cosine rule first and the sine rule afterwards, show clearly that sinθ = 1 14 H, proof

39 Question 38 (***+) 8cm O θ 8cm The figure above shows a circle with centre at O with radius 8 cm and a minor sector O, subtending an angle of θ radians at O. It is further given that the length of the circumference of the circle is twice plus 3 cm as large as the minor arc. a) Find the value of θ, in terms of π. b) Show that the area of the major sector O is 3( π + ) cm. θ = π

40 Question 39 (***+) π 3 O The figure above shows the design template of a car logo. The design consists of a circular ring of radius 6 cm enclosing a region, which is symmetrical about the line O. The angle O is 3 π. a) Find, to three significant figures, the perimeter of the shaded region of the logo. b) Show that the area of the shaded region is ( π ) cm perimeter 9.5 cm

41 Question 40 (***+) O 6 3 cm The figure above shows a circle with centre at O and radius 6 cm. The chord has length 6 3. a) Show that the angle O is π radians. 3 b) Show that the area of the minor segment, shown shaded in the figure above, is 3( 4π 3 3) cm. proof

42 Question 41 (***+) 6cm π 3 6cm 6cm π 3 6cm D The figure above shows two identical circles with centres at and, overlapping each other and meeting at the points and D. The radius of each circle is 6 cm. Each of the angles D and D is 3 π radians. The region, shown shaded in the figure above, enclosed by the two circles, including the overlap, is a car logo design. Find the area of the logo design. area 0 cm

43 Question 4 (***+) y ( 6,8) ( 8,6) O x The figure above shows a circular sector O with centre at the origin O. The points and have coordinates ( 6,8 ) and ( 8,6 ), respectively. a) Show that the angle O is approximately radians. b) Find, to decimal places, the area of the sector O. G, area 14.19

44 Question 43 (***+) 1 cm 1 cm θ 1 3 cm The figure above shows an isosceles triangle, where = 1 3 cm and = = 1 cm. The angle is θ radians. Two identical arcs centred at and are drawn inside the triangle. These arcs meet on. a) Show that π θ = radians. 6 b) Find the exact area of the triangle in terms of 3. c) Show that the area of the shaded region is given by ( π ) 18 3 cm. 36 3

45 Question 44 (***+) R 5 R 4 R 3 R R 1 O The figure above shows a circular sector O. The sides of the sector are equally divided into five equal parts. Using these divisions arcs are drawn inside the original sector, creating five distinct regions R 1, R, R 3, R 4 and R 5, as shown in the figure. Show that the areas of the regions R and R 5 are in the ratio 1:3. proof

46 Question 45 (***+) y cm R 5 x cm The figure above shows a triangle. The lengths of and are x cm and y cm, respectively. It is further given that 4 sin =, 5 8 sin = and sin =. 85 a) Show clearly that y = 1.7x. The area of the triangle is 1 cm. b) Find the value of x and the value of y. G, x = 5, y = 8.5

47 Question 46 (***+) ϕ 4cm 6cm 9cm θ D 5cm The figure above shows a trapezium D, where is parallel to D. The respective lengths of D, D, and D are 4 cm, 6 cm, 9 cm and 5 cm. The angle D is θ. a) Show clearly that 1 cosθ =. 3 b) Hence show further that sinθ = k The angle D is ϕ., where k is a fraction. c) Find the exact value of sinϕ. F, sin θ = 3, sinϕ = 1 6

48 Question 47 (***+) r cm c θ O r cm The figure above shows a circular sector O, centred at O. The radius of the sector is r cm and subtends an angle of θ radians at O. The area of the sector is 67.5 cm and its perimeter is 33 cm. y forming two suitable equations, or otherwise, determine the two possible pairs of values for r and θ. K, ( r θ ) = ( ) = 5 ( ), 7.5,.4 9, 3

49 Question 48 (***+) c m c 1.1 The figure above shows a river of constant width w metres with the points and located on one river bank and the point located on the other river bank. The distance is 85 metres. The angles and are 0.7 radians and 1.1 radians, respectively. Show that w is approximately 50 metres. K, proof

50 Question 49 (***+) O 3cm 3cm D 3cm 3cm R The figure above shows a circular sector O, of radius 6 cm, centred at O. The points and D are the midpoints of O and O, respectively. The triangle OD is equilateral. nother circular sector D, centred at D and of radius 3 cm, is drawn inside the circular sector O. The finite region R bounded by the circular arcs and, and the straight line segment, is shown shaded in the figure above. a) Show that the perimeter of R is ( 3 + 4π ) cm. b) Determine the exact area of R. 3π 9 3 4

51 Question 50 (***+) 6 cm 6 cm c 1.5 The figure above shows a circular sector of radius 6 cm subtending an angle 1.5 radians at. a) Find the perimeter and the area of the sector. different sector DEF has radius r cm and subtends an angle of θ radians at its centre D. E r cm F r cm c θ D b) Given that the two sectors have equal area but the perimeter of the sector DEF is 1.5 cm larger than the perimeter of the sector, determine the possible values of r and the corresponding values of θ. P = 19.5 cm, =.5 cm, ( ) ( c c r, θ = 7.5,0.8 ) or ( r, θ ) = ( 3,5 )

52 Question 51 (***+) The triangle has vertices at ( 5,), ( 3,0) and ( 1,6 ). The angle is denoted by θ. a) Use the cosine rule to show that cosθ = b) Hence, or otherwise, show that the area of the triangle is exactly 10. proof

53 Question 5 (***+) θ O ϕ 5 cm The figure above shows a circle of radius 5 cm, centred at O. The points, and lie on the circumference of the circle. The angles O and O are denoted by θ and ϕ, respectively. The sum of the areas of the sectors O and O is 0 cm. The length of the arc is 3.5 cm greater than the length of the arc. Determine the value of θ and the value of ϕ. c c θ = 1.15, ϕ = 0.45

54 Question 53 (***+) 60 6 cm 75 D 6 cm The figure above shows a triangle. The straight line D is such so that D = D = 6 cm. The angles D and D are 60 and 75, respectively. Find in appropriate degree of accuracy a) the length of D. b) the area of the triangle of D. c) the shortest distance from the vertex to the straight line. d) the length of. P, , 9 ( ) 1.9, ( ) , 10.6

55 Question 54 (***+) D E θ r cm The figure above shows a minor sector DE with radius r cm, subtending an angle of θ radians at. The sector is attached to a square D as shown forming a composite shape. The area and the perimeter of are 48cm and 8cm, respectively. y forming and solving two equations, find the value of r and the value of θ. r = 6, θ = 3

56 Question 55 (***+) 5cm θ 5cm 14cm R The figure above shows an isosceles triangle attached to a semicircle with as its diameter. It is further given that radians. = = 5 cm, = 14 cm and the angle is θ circular arc is drawn inside the semicircle, centred at with radius 5 cm. a) Determine the area of the triangle. b) Show that θ = radians, correct to three significant figures. c) Find the area of the region R, shown shaded in the figure. O, area of triangle = 168 cm, area of R cm

57 Question 56 (***+) O θ The figure above shows a quarter circle O with centre at O. The point lies on the curved part of the quarter circle so that the angle O is θ radians. Given that the length of the arc is four times as large as the length of the arc, π show that θ =. 10 proof

58 Question 57 (***+) circular sector has radius r cm and subtend an angle θ radians at its centre. The perimeter of the sector is 3 cm and its area is 15 Find the value of r and the value of θ. cm. V, c r = 10 cm, θ = 0.3

59 Question 58 (***+) O 6 3 cm P The figure above shows a circle with centre at O and radius 6 cm. The chord has length 6 3 cm. a) Show that the angle O is π radians. 3 The tangents to the circle at and meet at the point P. b) Show further that the area of the quadrilateral OP is 36 3 cm. c) Find the area of the shaded region bounded by the tangents and the circle. N, area = 1( 3 3 π )

60 Question 59 (***+) P 3a 4a a O θ D 3a 4a figure 1 Q figure Figure 1, shows a rectangle D where = D = 3a and D = = a. semicircle with diameter is attached to the rectangle. The rectangle and the semicircle are to be considered as a single composite shape X. Figure, shows a circular sector OPQ where OP = OQ = 4a. The sector has its centre at O, and POQ = θ radians. The sector is denoted as shape Y. a) Given that the area of X is equal to the area of Y, express θ in terms of π. b) Given further that the perimeter Y is greater than the perimeter of X, show that the difference between the perimeter of X and Y is 3 ( 4 4 a π ). π θ =

61 Question 60 (***+) 4 cm 6 cm 10 cm not drawn accurately 60 9 cm D The figure above shows a quadrilateral D, with side lengths,, D and D are 6 cm, 4 cm, 10 cm and 9 cm, respectively. The angle D is 60. a) Show that D is 3 7 cm. b) Find, to one decimal place, the size of the angle D. c) Determine, to one decimal place, the area of the quadrilateral D.., area 9.4

62 Question 61 (***+) D 6cm θ 6cm not drawn accurately The figure above shows a sector D, of radius 6 cm and angle θ radians. Given that the area of the triangle and the area of segment D are in the ratio 4 :1, show that 8θ 5sin θ = 0. proof

63 Question 6 (***+) O r a π 6 a r not drawn accurately M N The figure above shows a circular sector O. The sector has radius r cm and subtends an angle of 6 π at O. The straight line through M and N is such so that OM = ON = a cm. Given that the straight line through M and N divides the sector into two regions of equal area, show that π a = r. 6 J, proof

64 Question 63 (***+) l 6 cm P not drawn accurately Q R The figure above shows a circle of radius 6 cm, centre at point. The line l is the tangent to the circle at the point P and the point R lies on l. The line segment R meets the circle at the point Q. Given that the length of the arc PQ is π cm, show that the area of the shaded region is given by 6( 3 3 π ). proof

65 Question 64 (***+) The triangle has = 13 cm and = 15 cm. Given that = 60, determine the possible values of. = 7 cm or 8 cm

66 Question 65 (****) D 60 E D is a trapezium where is parallel to D. The angle is 60 and = 90. The side is extended from to E so that ED = 90, as shown in the figure above. It is further given that E = 15 and ED = 60. a) Find, correct to 1 decimal place, the value of and the value of D. b) alculate, correct to 1 decimal place, the angle D. 81.6, D 49.6, D 44.0

67 Question 66 (****) 7cm O 7cm c 1.8 not drawn accurately The figure above shows a circle with centre at O and radius 7 cm. The points and lie on the circle so that the angle O is 1.8 radians. The tangents to the circle at the points and meet at the point. The region shown shaded in the figure above, is enclosed by the two tangents and, and the circle. Determine the area of this region. area 17.6 cm

68 Question 67 (****) STGE D 5m 1m STGE 0m O 1m D 5m O The two diagrams above show an orchestral stage D which is part of a circular sector O, centred at O and of radius 17 m. The points and D lie on O and O respectively so that O = OD = 1 m and D = 0 m. a) Show that O = 1.97, correct to three significant figures. b) alculate the area of the stage. There are 4 rows of seats with their backs arranged in concentric circles, centred at O. The radii of these circles are 1 m, 13.1 m, 14. m and 15.3 m. c) Given further that each seat requires a length of 83 cm along the arc, find approximately how many more seats are in the back row than in the front row. M, area 18m, an extra 17 seats

69 Question 68 (****) not drawn accurately D π 6 1 O π 4 The figure below shows a right angled triangle O where O = π. 4 O = π and nother triangle OD is drawn next to the triangle O, so that DO is a straight line, DO = 1 units and DO = π. 6 Finally a circular sector O is drawn, centred at O, with radius O, so that DO is a straight line. a) Show that the length of O is 3.18 units, correct to two decimal places. b) Find the area of the sector O. c) Hence show that the area of the shaded region is approximately 1 square units. 11.0

70 Question 69 (****) r O not drawn accurately r The figure above shows a circular sector O of radius r, centred at O, with perimeter of 60 units. The area of the sector is denoted by. a) Show clearly that = 30r r. The value of r can vary but the perimeter of the sector is fixed. b) y completing the square, or otherwise, find the maximum value of and the value of r which produces this maximum value for. max = 5, r = 15

71 Question 70 (****) θ O D not drawn accurately The figure above shows a model of the region used by shot putters in to throw the shot. The throwing region consists of a minor circular sector O of radius 1 3 metres subtending an angle θ radians at O. The chord is 36 metres. The shot putter s region OD is a major circular sector of radius 3 3 metres, where and D lie on O and O, respectively. π a) Show that θ =. 3 b) Find, in terms of π, the total area of throwing region and shot putter s region. c) Show that the total perimeter of throwing region and shot putter s region, shown shaded in the figure above, is 6( π + 3) 3. 16π

72 Question 71 (****) not drawn accurately 1cm R 3 10 π O The figure above shows a circular arc O of radius 1 cm, subtending an angle of 3π radians at O. 10 Find to three significant figures a) the length of the arc. b) the area of the sector O. The point lies on O so that O =. The region R, shown shaded in the figure, is bounded by the arc and the straight lines and. c) Determine, to three significant figures, the perimeter and area of R cm, 67.9 cm, 3.3 cm, cm

73 Question 7 (****) P θ 1 cm 9.8 cm 5.7 cm 10 cm Q 0 R figure 1 figure Figure 1 shows the triangle, where = 1 cm, = 10 cm and = θ so that cosθ = 5. 9 a) Use the cosine rule to form a suitable quadratic, and hence show that one of the two possible values for the length of is 6 cm and find the other. Figure shows a different triangle PQR, where PQ = 9.8 cm, PR = 5.7 PQR = 0. cm and b) Use the sine rule to find, to the nearest degree, the two possible values of QPR. L, = 7.33 cm, QPR = 16 or 14 3

74 Question 73 (****) not drawn accurately S 1 S r O θ The figure above shows a semicircle of radius r cm, where O is a diameter with point O the centre of the semicircle. The point lies on the circular part of the semicircle so that the angle O is θ radians. The chords and define two segments S 1 and S, respectively. Given that the area of S 1 is four times as large as the area of S, show that π + 3sinθ = 5θ. proof

75 Question 74 (****) 8cm O r not drawn accurately π 3 8cm The figure above shows a sector of radius 8 cm, centred at and subtending an angle of 3 π radians at. circle centred at O and of radius r cm is inscribed in the sector. a) Show clearly that 8 r =. 3 b) Find in terms of π, the area of the shaded region in the figure. area = 3 9 π

76 Question 75 (****) 9cm not drawn accurately 1cm 4cm 3 D The figure below shows the quadrilateral D where is 9 cm, is 1 cm and D is 4 cm. The angle is 90 and the angle D is 3. Find, to three significant figures, the area of the quadrilateral D 345

77 Question 76 (****) 1cm not drawn accurately 6cm D E The figure above shows the rectangle D where is 6 cm and is 1 cm. n arc of a circle with centre at and radius 1 cm is drawn inside the quadrilateral, meeting the side D at the point E. Find the area of the shaded region E. area = 7 1π

78 Question 77 (****) 5cm not drawn accurately The figure above shows the design for an earring. The design consists of a part of a circle of radius 3 cm centred at and another part of a circle of radius 4 cm centred at. The circles overlap in such a way so that the distance is 5 cm. Find, to three significant figures, the perimeter of the design. perimeter 33.3

79 Question 78 (****) x cm 37 cm 60 y cm not drawn accurately The figure above shows a triangle where is x cm, is y cm and is 37 cm. The angle is 60. Given further that the area of the triangle is 7 3 cm, determine by solving two simultaneous equations the value of x and the value of y. V, x = 37

80 Question 79 (****) The figure above shows a curved triangle, known as a Reuleaux triangle, which is constructed as follows. Starting with an equilateral triangle of side length cm, a circular arc is drawn with centre at. Two more circular arcs and are drawn with respective centres at and. Show that the area of this Reuleaux triangle is ( π 3) cm. proof

81 Question 80 (****) P Q 1 c θ O The figure above shows a grid used to help spectators estimate the throwing distances of athletes in a shot put competition. The grid consists of circular sectors each with centre at O and the angle POQ is θ radians. The radius of the smaller sector is 10 metres and each of the other sectors has a radius metres more than the previous one. a) Show that i. the perimeter of the shaded region, labelled 4 is ( 4 30θ ) + metres. ii. the perimeters of the regions, 3, 4,..., 7 are terms of an arithmetic series. The perimeter of 4 is 1.4 times larger than the perimeter of 1. b) Determine the value of θ. c θ = 1.5

82 Question 81 (****+) O 6 3 cm The figure above shows an equilateral triangle circumscribed by a circle of radius 6, with centre at O. The circular segment, shown shaded region in the figure above, is bounded by the straight line. Show that the area of the segment is 3( 4π 3 3) cm. proof

83 Question 8 (****+) not drawn accurately D π 6 1 O π 4 The figure above shows a triangle O with O = π and O = π. 4 nother triangle OD is drawn next to the triangle O, so that DO is a straight line, DO = 1 units and DO = π. 6 Finally a circular sector O is drawn, centred at O, with radius O, so that DO is a straight line. a) Show that the length of O is 6( 6 + ). b) Find the exact area of the sector O. c) Hence show that the area of the shaded region is ( )( π ) ( + 3) π

84 Question 83 (****+) 7 cm O 4 cm The figure above shows a circle with centre at O and radius r. The straight line is a chord to the circle. The perpendicular bisector of passes through O and meets the circle at the point, as shown in the figure. Given that determine the value of r. = 4 cm and the length of the perpendicular bisector is 7 cm, r = 37

85 Question 84 (****+) 6cm O not drawn accurately 6cm D The figure above shows a circle of radius 6 cm, centred at O. n arc O with centre at and radius 6 cm is drawn inside the circle. second arc OD is drawn with centre at and radius 6 cm. Show clearly that the area of the shaded region OD is ( π ) cm. proof

86 Question 85 (****+) y not drawn accurately O D x The figure above shows a circle with centre at ( 3,6). The points ( 1,5 ) ( p, q ) lie on the circle. The straight lines D and D are tangents to the circle. The kite D is symmetrical about the straight line with equation x = 3. a) alculate the radius of the circle. b) State the value of p and the value of q. and c) Find an equation of the tangent to the circle at. d) Show that the angle is approximately.14 radians. e) Hence determine, to three significant figures, the area of the shaded region bounded by the circle and its tangents at and. U, r = 5, p = q = 5, y = 7 x, area 4.46

87 Question 86 (****+) r 3 π 1 D r 1 R E The figure above is constructed as follows. ED is a circular sector with centre at and radius 1 units, subtending an angle of π radians at. 3 E is a quarter circle with centre at and radius r units, so that D is a straight line and E is a right angle. The shaded region R is bounded by the arcs ED and E, and the straight line D. Show that the area of R is 3( 7π + 6 3) square units. Z, proof

88 Question 87 (****+) The figure below shows a square D with side length of 6 cm. circular arc D is drawn inside the square with centre at and radius of 6 cm. nother circular arc D is drawn inside the square with centre at and radius of 6 cm also, so that the two arcs bound a finite area, shown shaded in the figure above. D 6cm Show that area of the shaded region is 18( π ) cm. proof

89 Question 88 (****+) The distance between the town of rundel ( ) and the town of erry( ) is 60 km. erry is on bearing of 75 from rundel. The village of rake ( ) is on a bearing of 10 from rundel and on a bearing of 195 from erry. The village of Dorking ( D ) is on a bearing of 135 from rundel and on a bearing of 10 from erry. a) Find, to three significant figures where appropriate, the distance between i. erry and rake. ii. erry and Dorking. iii. rake and Dorking. b) State the bearing of Dorking from rake km, 53.8 km, 16.1 km, 55

90 Question 89 (****+) The figure below shows a circle with centre at O and radius r. The points and lie on the circle so that the angle O is θ radians. r O θ r not drawn accurately The chord divides the circle into a major segment and a minor segment. Given that the area of the major segment is 4 times as large as the area of the minor segment, show clearly that 5θ 5sinθ = π. proof

91 Question 90 (****+) The figure below shows a circle with centre at O and radius r. The points and lie on the circle so that the angle O is θ radians. r O θ r not drawn accurately Given that the length of the arc is 1.5 times as large as the chord, show clearly that area of the sector O 3 =. area of the triangle O cosθ You may use the fact that sin θ sinθ cosθ. proof

92 Question 91 (****+) 4cm P Q 5cm R 4cm The figure above shows a triangle where = 90. The lengths of, and are 3 cm, 4 cm and 4 cm, respectively. Three arcs are drawn inside the triangle with centres the three vertices of the triangle. The arcs so that they touch each other in pairs at the points P, Q and R. Find the area of the shaded region, correct to three significant figures. area = cm

93 Question 9 (****+) The figure below shows a square D of side length a cm, circumscribed by a circle. Four semicircles are then drawn outside the square having each of the sides of the square as a diameter. a a lunes D Each of the four regions bounded by a semicircle and the circumscribing circle is known by the mathematical name of a lune, i.e. moon shaped. Show that the area of the four lunes is equal to the area of the square D. proof

94 Question 93 (****+) The figure below shows the plan of three identical circular cylinders of radius 6 cm, held together by an elastic band. Show that the exact length of the stretched elastic band is 1( π + 3) cm. proof

95 Question 94 (****+) The figure below shows the plan of three identical circular cylinders of radius 6 cm, that fit snugly inside a larger cylinder. Show that the radius of the larger cylinder is cm. proof

96 Question 95 (****+) x + 3 F 15 y D x E 10 The figure above shows the triangle. The point D lies on so that the straight line D meets at right angles. The point E lies on and the point F lies on, so that the straight line DF is parallel to and the straight line EF is parallel to D. It is further given that the lengths, in cm, of E, F, DE, F and D are 10, 15, x, x + 3 and y, respectively. a) Determine the value of x. b) Show clearly that y = 9.6. c) Find, correct to three significant figures, the area of the triangle. x = 6, area = 9

97 Question 96 (****+) The following information is known about 4 coplanar points. 1. is north east of.. is on a bearing of 075 from. 3. is on a bearing of 85 from. 4. D is south west of. 5. = D = 36. Determine, correct to decimal places the bearing of from D. SP-Q, 37.33

98 Question 97 (****+) The island state of Trigland has declared an exclusive economic zone into the sea, which is within 6 miles from every point of its coastline. The island of Trigland is a rectilinear triangle of sides 13, 14 and 15 miles. Determine, in exact form, the total economic zone of Trigland, which consists of land and sea. SP-, π

99 Question 98 (*****) 3 cm P Q 3 cm R D The figure above shows, two identical circles, centred at P and Q, and a third circle, centred at R, are touching each other externally. The three circles fit snugly inside a square D, of side length 3 cm, so that PQ is parallel to. Determine the size of radius of the circle centred at R. r = 9 cm

100 Question 99 (*****) The figure below was constructed as follows. D is a square with side length 6 cm. Two quarter circles are drawn inside the square, each of radius 6 cm, with centres at the points and D. 6 cm D Show that the area of the shaded region is ( ) π cm. T, proof

101 Question 100 (*****) D The figure above shows a square D of side length units. The vertices and lie on the circumference of a circle while the side D is a tangent to the same circle. Determine the radius of this circle. r = 5 = 1.5 4

102 Question 101 (*****) The figure below was constructed as follows. D is a square with side length 8 cm. Four identical quarter circles are drawn inside the square whose centres are located at each of the four corners of the square. The radii of the quarter circles are such so that the four quarter circles meet at the centre of the square. 8 cm D Show that the area of the shaded region is 3( π ) cm. T, proof

103 Question 10 (*****) (non calculator) O π π The figure above shows a circle with centre at O. The straight line segment is a tangent to the circle at, so that the angle O is 1 3 π radians. Determine the radius of the circle given further that the area of the shaded region in the cm. figure is ( 6 3 π ) r =

104 Question 103 (*****) (non calculator) 7 cm 8 cm 5 cm The figure above shows the triangle where is 7 cm, is 5 cm and is 8 cm. Show that the exact area of this triangle is 10 3 cm. proof

105 Question 104 (*****) 75 1 m 60 The figure above shows a triangle. The length of is 1 m. The angles and are 75 and 60, respectively. The height of the triangle from the vertex to the side is h cm. Show that tan 75 tan 60 h = tan 75 + tan 60. SP-T, proof

106 Question 105 (*****) (non calculator) R x x +10 D The figure above shows a right angled triangle, where the angle is 30. The point D lies on so that the angle D is 75. The length of D is x cm and he length of D is x + 10 cm. Show that the length of is ( ) [you may assume that tan 75 = + 3 ] proof

107 Question 106 (*****) (non calculator) D E 4 cm The figure above was constructed as follows. semicircle with diameter of cm is first drawn. nother semicircle is circumscribed by the first semicircle so that its diameter D is parallel to. Show that the area of the shaded region is ( π ) cm. proof

108 Question 107 (*****) (non calculator) c 0.9 O R 6cm The figure above shows a circular arc O of radius 6 cm, subtending an angle of 0.9 radians at O. The point lies on O so that O =. The region R, shown shaded in the figure, is bounded by the arc and the straight lines and. Determine the perimeter of R cm

109 Question 108 (*****) (non calculator) 60 6 cm 75 D 6 cm The figure above shows a triangle. The line D is such so that and 75, respectively. D = D = 6 cm and the angles D and D are 60 Show that a) The shortest distance from the vertex to the side is b) The length squared is ( ) proof

110 Question 109 (*****) D R The figure above shows a straight line intersecting a circle at the points and so that = 8 units. nother straight line intersects the same circle at the points and D so that D = 1 units. The two straight lines intersect each other at right angles at the point R. Given further that D = 4 units, determine the length of the radius of the circle. r = 10

111 Question 110 (*****) ψ D ϕ θ The point D lies inside the triangle, so that D = D and D = 1 π. Let θ = D, ϕ = D and ψ = D. Show that sinψ = sinθ sinϕ. SP-I, proof

112 Question 111 (*****) Two coplanar circles, with respective radii and 6, intersect each other at the points and. The tangent to one of the circles at, intersects the tangent to the other circle at at right angles. Show that the total area enclosed by the two circles is ( ) 1 19 π SP-L, proof

113 Question 11 (*****) hiker on a mountain walk has injured himself. He rings the rescue station which is located at the point with coordinates (,1 ). He reports that he is lying injured by a river bank where he can see a ruined tower, which his compass indicates that it is located South-West from his position. It is known to the rescue station that the only river in the area has equation x = 8 and,3 on the coordinate axes. the ruined tower is located at the point with coordinates ( ) The rescuers set off immediately from the Rescue Station and travel directly towards the hiker. When the rescuers are half-way into their journey, the hiker rings again. He says that he made a mistake in reading his compass and the ruined tower is in fact located North-West from his position. The rescuers turn and head directly towards the true location of the hiker. alculate the angle, as a bearing, at which the rescuers are heading after the hikers second phone call. Give the answer in the form µ π + arctan λ, where µ and λ are constants to be found. SP-P, 1 π + arctan 8 3

114 Question 113 (*****) Two circles of different radii are touching each other externally. The two circles are enclosed by a square so that all 4 sides of the squares are tangents to the circles, as shown in the figure above. Given that the radius of the smaller circle is r and the radius of the larger circle is r, determine the exact area of the square in terms of r. SP-O, area = 9 ( 3 + ) r

115 Question 114 (*****) θ a The figure above shows an isosceles triangle, where The side has length a and the angle is θ. Show that the area of the triangle is =. 1 tan 4 a θ. S, proof

116 Question 115 (*****) 10 cm x cm 1 cm 6 cm θ D ϕ cm The figure above shows the triangle, where = 10 cm, = 1 cm and = 8 cm. The point D lies on so that D = 6 cm, D = D cm and = x cm. The angle D is denoted by θ and the angle D is denoted by ϕ. a) Express cosθ and cosϕ in terms of x. b) Use part (a) to find the length of D. c) Hence show that the area of the triangle D is exactly cm. SP-Y, proof

117 Question 116 (*****) The triangle is such so that = 90 and = 6 cm. The point P lies on and the point Q lies on in such a way so that PQ = 90 and P = Q = 1 cm. Show that the straight line segment P is exactly 7 cm. T, proof

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