REVISION EXERCISES ON GEOMETRY END TRIGONOMETRY 1 The bearing of B from A is 030. Find the bearing of A from B. 2 For triangle ABC, AB = 60 cm, BC = 80 cm and the magnitude of angle ABC is 120. Find the length of AC, correct to two decimal places. 3 The angle of depression from a point A to a ship at B is 10. If the distance, BX from B to the foot of the cliff at X is 800 m, find the height of the cliff, to the nearest metre.
4 AB = 8 cm, BC = 6 cm and AC = 12 cm. Find the magnitude of each of the angles of triangle ABC. B A C 5 Find the value of x if AC = 8 cm and A C = 12, A A= x cm and AB = x 2. C C A A B 6 A ship sails on a bearing of 028 o for 20 km until it reaches a point A. It then sails on a bearing of 330 o until it is due north of O. How far is it from O. N A O
7 In parallelogram ABCD, AB = 10 cm and BC = 7 cm. Find the magnitude of each of the angles of the parallelogram. D C A B 8 Find the length of the hypotenuse of triangle ABC (correct to two decimal places), where AB = 6 cm and AC = 2 cm.. C 2 cm A B 6 cm 9 State the values of x and y giving reasons for your answers. y o 95 o 40 o x o 10 In triangle ABC, AB = 6 cm, BC = 12 cm and AC = 15 cm. Find the magnitude of BAC correct to one decimal place.
6 cm B 12 cm A 15 cm C 11 In triangle ABC, AP = x cm, PB = 12 cm, BQ = 16 cm and QC = 8 cm. PQ is parallel to AC. Find the value of x. B 12 cm 16 cm P Q x cm 8 cm A C 12 In a parallelogram ABCD, AB = CD = 6 cm and BC = AD = 12 cm. If BCD = 62 o, find the length of the diagonal AC to the nearest centimetre. 13 A boat sails from point A on a bearing of 038 0 for 3000 m to point C. The boat then alters course to a bearing of 318 o and after sailing for 3300 m it reaches a point B.
B North C A a Show that the magnitude of angle ACB is 100 o. (Suitable calculation must be shown.) b Find the distance AB correct to the nearest metre. c Find the size of angle BAC correct to one decimal place. d Find the bearing of B from A
14 For the diagram shown below find the size of acute angle ABC to the nearest degree. B 9 m C 13 m 15 o 78 o D 26 o A
15 A surveyor wishes to establish the width of a river, as shown in the diagram below. On the far side are two landmarks, a tree and a tower which are known to be 1600 m apart. The surveyor walks in a straight line parallel to the line TB. He notes positions A and C where AXB and CXT are straight lines and X is a point on the near side of the river. a Find d (using Pythagoras Theorem), correct to the nearest metre. b Find w (using similar triangles), correct to the nearest metre. c Find magnitudes of XAC and XCA, correct to one decimal place. d Give the bearing of: i B from A ii T from C
16 A ramp inclined at 20 to the horizontal has a vertical pole TX placed on it. Wires TA, TB and TC support it. The angle TBX is recorded as 36 at a point 10 m from the base of the pole X. AB = 12 m. This is shown on the diagram below. a Find TX, correct to one decimal place. b Find TX, correct to one decimal place, where X is a point vertically below X. c A canvas sail XTC is placed at the top of the ramp as shown. X Y = 5 m. Find: i XC, correct to one decimal place ii the area of the sail, correct to the nearest square metre. d Find the length of wire TA. e Find the magnitude of angle TAB.
17 ABCDE is a regular pentagon. O is the centre of the pentagon with OA = OB = OC = OD = OE Each side length of the pentagon is 1 cm. a Find the magnitude of: i EOD ii OED iii AED b Find the distance: i OA, correct to four decimal places ii AC, correct to four decimal places c Find the circumference of the circle with centre O and radius OA, correct to three decimal places. (This circle is called the circumcircle of the pentagon.) d Find the area of the pentagon, correct to three decimal places. e Find the area of the circle with centre O and radius OA, correct to three decimal places. f Find the area of the shaded region, correct to one decimal place
g A right pyramid has a base ABCDE as shown below. V is 5 cm vertically above O. V B 1 cm A 1 cm 5 cm 1 cm 1 cm E C 1 cm D i Find the length of VE, correct to one decimal place. ii Find the magnitude of EVD, correct to one decimal place. iii Find the surface area of the pyramid, correct to the nearest square cm. h The volume of a right pyramid is given by the formula V = 1/3Ah, where A is the area of the base and h is the height of the pyramid. Find, correct to one decimal place, the volume of: i the pentagonal pyramid shown in part g ii the right cone with a base which is the circumcircle of the pentagon (see diagram below) and height 5 cm. iii Find the difference between the volumes of the two solids. i The pyramid and cone in part h are scale models for a large structure. The scale is 1:500. Find the volume of the cone and pyramid of the larger structure.
Answers 1 a 210 2 121.66 cm 3 141 m 4 BAC = 26.38 ; ABC = 117.28 ; BCA = 36.34 5 x = 2 6 3392 km 7 ADC = 109.6 ; DCB = 70.4 ; CBA = 109.6 ; DAB = 70.4 8 6.32 cm 9 x = 40 ; y = 95 10 49.5 11 x = 6 12 16 cm 13 a 38 + ACB + 62 = 180 ACB = 100 b 4830 m c 42.3 d 356 14 57 15 a 274 m b 877 m c 54.6 ; 41.9 d i 035 ii 312 16 a 10.5 m b 18.0 m c i 5.3 m ii 26 m 2 d 27.4 m e 21.1 17 a i 72 ii 54 iii 108
b i 0.8507 cm ii 1.6180 cm c 5.345 cm d 1.721 cm 2 e 2.273 cm 2 f 0.6 cm 2 g i 5.1 cm ii 11.3 iii 14 cm 2 h i 2.9 cm 3 ii 3.8 cm 3 iii 0.9 cm 3 i Cone: 475 m 3 ; Pyramid: 360 m 3 ; calculated using values in h i and h ii