ABBASI MOHAMMED ASIM Page: 1 Ch15-Trigonometry. Answer.. m [2] Answer (a).. cm [2] Answer (b).. cm 2 [1] NOT TO SCALE

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1 h15-trigonometry 1. entrance h m 3.17 m 5 pavem ent shop has a wheelchair ramp to its entrance from the pavement. The ramp is 3.17 metres long and is inclined at 5 to the horizontal. alculate the height, h metres, of the entrance above the pavement. Show all your working. nswer.. m 2. square, of side 8 cm, has another square, PQRS, drawn inside it. P, Q, R and S are at the midpoints of each side of the square, as shown in the diagram. P S Q R (a) alculate the length of PQ. nswer (a).. cm (b) alculate the area of the square PQRS. nswer (b).. cm 2 SI MOHMME SIM Page: 1 mdasimabbasi@yahoo.co.in

2 3. plane flies from uckland () to Gisborne (G) on a bearing of 115. The plane then flies on to Wellington (W). ngle GW = 63. N orth 115 N orth 410 km 63 G 400 km W (a) alculate the bearing of Wellington from Gisborne. nswer (a)... (b) The distance from Wellington to Gisborne is 400 kilometres. The distance from uckland to Wellington is 410 kilometres. alculate the bearing of Wellington from uckland. nswer (b)... [4] 4. E O,, and lie on a circle, centre O, radius 8 cm. and are tangents to a circle, centre O, radius 4 cm. is a rectangle. (a) alculate the distance E. nswer (a) E =. cm (b) alculate the shaded area. nswer (b). cm 2 SI MOHMME SIM Page: 2 mdasimabbasi@yahoo.co.in

3 5. In triangle, = 6 cm, = 8 cm and = 12 cm. ngle = alculate the area of the triangle. 6 cm 8 cm cm nswer cm x cm 150 cm 24x cm The right-angled triangle in the diagram has sides of length 7x cm, 24x cm and 150 cm. (a) Show that x 2 = 36 (b) alculate the perimeter of the triangle. nswer (b). cm 7. 13x cm 5x cm 17x cm 12x cm y is a trapezium. (a) Find the area of the trapezium in terms of x and simplify your answer. nswer (a). cm 2 (b) ngle = y. alculate the value of y. nswer (b) y =.. SI MOHMME SIM Page: 3 mdasimabbasi@yahoo.co.in

4 8. 70 The diagram shows three touching circles. is the centre of a circle of radius x centimetres. and are the centres of circles of radius 3.8 centimetres. ngle = 70. Find the value of x. nswer x = cm 7 cm 12 cm The diagram shows a trapezium. = 12 cm, = 9 cm and the perpendicular distance between these parallel sides is 7 cm. =. (a) pproximately halfway down your page, draw a line of length 12 cm. (b) Using a straight edge and compasses only, construct the perpendicular bisector of. (c) omplete an accurate drawing of the trapezium. (d) Measure angle, giving your answer correct to the nearest degree. SI MOHMME SIM Page: 4 mdasimabbasi@yahoo.co.in

5 (e) Use trigonometry to calculate angle. Show all your working and give your answer correct to 1 decimal place. (f) On your diagram, (i) draw the locus of points inside the trapezium which are 5 cm from, (ii) using a straight edge and compasses only, construct the locus of points equidistant from and from, (iii) shade the region inside the trapezium containing points which are less than 5 cm from and nearer to than to. 10. d c E O OE is a regular hexagon. With O as origin the position vector of is c and the position vector of is d. (a) Find, in terms of c and d, (i), (ii) OE, (iii) the position vector of. (b) The sides of the hexagon are each of length 8 cm. alculate (i) the size of angle, (ii) the area of triangle, (iii) the length of the straight line, (iv) the area of the hexagon. SI MOHMME SIM Page: 5 mdasimabbasi@yahoo.co.in

6 11. l 0.7 cm h The diagram shows a pencil of length 18 cm. It is made from a cylinder and a cone. The cylinder has diameter 0.7 cm and length 16.5 cm. The cone has diameter 0.7 cm and length 1.5 cm. (a) alculate the volume of the pencil cm 1.5 cm [The volume, V, of a cone of radius r and height h is given by V = 1 r 2 h. 3 (b) x cm 18 cm w cm Twelve of these pencils just fit into a rectangular box of length 18 cm, width w cm and height x cm. The pencils are in 2 rows of 6 as shown in the diagram. (i) Write down the values of w and x. (ii) alculate the volume of the box. (iii) alculate the percentage of the volume of the box occupied by the pencils. (c) Showing all your working, calculate (i) the slant height, l, of the cone, (ii) the total surface area of one pencil, giving your answer correct to 3 significant figures. [The curved surface area,, of a cone of radius r and slant height l is given by = πrl.] [6] SI MOHMME SIM Page: 6 mdasimabbasi@yahoo.co.in

7 12. N orth S 7 km 30 P km R 14 km Q The quadrilateral PQRS shows the boundary of a forest. straight 15 kilometre road goes due East from P to R. (a) The bearing of S from P is 030 and PS = 7 km. (i) Write down the size of angle SPR. (ii) alculate the length of RS. [4] (b) ngle RPQ = 55 and QR = 14 km. (i) Write down the bearing of Q from P. (ii) alculate the acute angle PQR. (iii) alculate the length of PQ. (c) alculate the area of the forest, correct to the nearest square kilometre. [4] SI MOHMME SIM Page: 7 mdasimabbasi@yahoo.co.in

8 13. P 3 cm 5 cm F M 6 cm The diagram shows a pyramid on a rectangular base, with = 6 cm and = 5 cm. The diagonals and intersect at F. The vertical height FP = 3 cm. (a) How many planes of symmetry does the pyramid have? (b) alculate the volume of the pyramid. 1 [The volume of a pyramid is area of base height.] 3 (c) The mid-point of is M. alculate the angle between PM and the base. (d) alculate the angle between P and the base. [4] (e) alculate the length of P. 14. P 13 cm E 6 cm 8 cm The diagram shows a pyramid on a horizontal rectangular base. The diagonals of meet at E. P is vertically above E. = 8 cm, = 6 cm and P = 13 cm. SI MOHMME SIM Page: 8 mdasimabbasi@yahoo.co.in

9 (a) alculate PE, the height of the pyramid. (b) alculate the volume of the pyramid. [The volume of a pyramid is given by 1 3 area of base height.] (c) alculate angle P. (d) M is the mid-point of and N is the mid-point of. alculate angle MPN. (e) (i) alculate angle P. (ii) K lies on P so that K = 4 cm. alculate the length of K y 1 y y + 2 The diagram shows a right-angled triangle. The lengths of the sides are given in terms of y. (i) Show that 2y 2 8y 3 = 0. (ii) Solve the equation 2y 2 8y 3 = 0, giving your answers to 2 decimal places. (iii) alculate the area of the triangle. [4] SI MOHMME SIM Page: 9 mdasimabbasi@yahoo.co.in

10 16. N orth L W km 1600 km H 95 J The diagram shows the positions of four cities in frica, Windhoek (W), Johannesburg (J), Harari (H) and Lusaka (L). WL = 1400 km and WH = 1600 km. ngle LWH = 13, angle HWJ = 36 and angle WJH = 95. (a) alculate the distance LH. [4] (b) alculate the distance WJ. [4] (c) alculate the area of quadrilateral WJHL. (d) The bearing of Lusaka from Windhoek is 060. alculate the bearing of (i) Harari from Windhoek, (ii) Windhoek from Johannesburg. (e) On a map the distance between Windhoek and Harari is 8 cm. alculate the scale of the map in the form 1 : n. SI MOHMME SIM Page: 10 mdasimabbasi@yahoo.co.in

11 cm 92 X 26.8 cm 20.1 cm,, and lie on a circle. and intersect at X. ngle X = 55 and angle X = 92. X = 26.8 cm, X = 40.3 cm and X = 20.1 cm. (i) alculate the area of triangle X You must show your working. (ii) alculate the length of. You must show your working. (iii) Write down the size of angle. Give a reason for your answer. (iv) Find the size of angle. (v) Write down the geometrical word which completes the statement Triangle X is to triangle X. (vi) alculate the length of X. You must show your working. SI MOHMME SIM Page: 11 mdasimabbasi@yahoo.co.in

12 18. E 25 cm M O 14 cm O is a rhombus with sides of 25 cm. The length of the diagonal O is 14 cm. (a) Show, by calculation, that the length of the diagonal is 48 cm. (b) alculate, correct to the nearest degree, (i) angle, (ii) angle O. (c) = 2p and O = 2q. Find, in terms of p and q, (i) O, (ii) O. (d) E is parallel to O and E is a straight line. Find, in its simplest form, OE in terms of p and q. (e) M is the mid-point of E. Find, in its simplest form, OM in terms of p and q. SI MOHMME SIM Page: 12 mdasimabbasi@yahoo.co.in

13 (f) O is the origin of a co-ordinate grid. O lies along the x-axis and q = ( is vertical and = 48.) Write down as column vectors 7. 0 (i) p, (ii). (g) Write down the value of E. 19. S 8 cm P 6 cm N R 7 cm M Q The diagram above shows the net of a pyramid. The base is a rectangle 8 cm by 6 cm. ll the sloping edges of the pyramid are of length 7 cm. M is the mid-point of and N is the mid-point of. (a) alculate the length of (i) QM, (ii) RN. (b) alculate the surface area of the pyramid. SI MOHMME SIM Page: 13 mdasimabbasi@yahoo.co.in

14 (c) P 7 cm G H X N 6 cm M 8 cm The net is made into a pyramid, with P, Q, R and S meeting at P. The mid-point of is G and the mid-point of is H. The mid-point of is G and the mid-point of is H. The diagonals of the rectangle meet at x. (i) Show that the height, PX, of the pyramid is 4.90 cm, correct to 2 decimal places. (ii) alculate angle PNX. (iii) alculate angle HPN. (iv) alculate the angle between the edge P and the base. (v) Write down the vertices of a triangle which is a plane of symmetry of the pyramid. SI MOHMME SIM Page: 14 mdasimabbasi@yahoo.co.in

15 m 30 m 80º 26 m The diagram shows the plan of a garden. The garden is a trapezium with = 26 metres, = 18 metres and angle = 80. straight path from to has a length of 30 metres. (a) (i) Using a scale of 1: 200, draw an accurate plan of the garden. (ii) Measure and write down the size of angle and the size of angle. (iii) second path is such that all points on it are equidistant from and from. Using a straight edge and compasses only, construct this path on your plan. (iv) third path is such that all points on it are equidistant from and from. Using a straight edge and compasses only, construct this path on your plan. (v) In the garden, vegetables are grown in the region which is nearer to than to and nearer to than to. Shade this region on your plan. b) Use trigonometry, showing all your working, to calculate (i) angle, (ii) the length of, [4] (iii) the area of the garden. SI MOHMME SIM Page: 15 mdasimabbasi@yahoo.co.in

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