INTERMEDIATE MATHEMATICS (STEM C) 95 Answers Exercise 2.1 1. a) 71.62 b) 45.84 c) 90 d) 300 2. a) 0.4712 rad b) 2.377 rad c) 3.578 rad d) 1.478 rad 3. a) π 36 13 b) π 9 5 c) π 9 13 Exercise 2.2 1. 4.088 rad 2. 2.46 rad 3. OP = 5 cm, OQ = 8 cm 4. 6 cm 5. 3 cm Exercise 2.3 1. a) 1.6667 rad b) 95.51 c)120cm2 2.a) 40.68 b) 4.615cm 3. a) 7.774cm b) 27.472cm2 c) 27.5cm 4. 1.864cm2 5. a) π 3 2 b) 40.47cm 6. 1.06rad/ 60 7. 7. a) 8.73cm b) 43.64cm2 c) 65.22cm2 8. r=16.53cm and A= 16.46cm2 9. 9.81cm 10. 0.633 rad 11. a) 0.927rad b) 10.635cm 12. 77.07 cm2 13. a) 6.84cm b) 12.49cm c) 27.25cm
INTERMEDIATE MATHEMATICS (STEM C) 96CHAPTER 5 SOLUTION OF TRIANGLESObjectives At the end of this chapter the student should be able to Use the Pythagoras’ Theorem understand and solve problems using the concept of sine rule, cosine ruleandarea of triangles. 5.1 Right-Angle Triangles Solution The Pythagoras’s Theorem and the trigonometric ratio is used to find the angles andsides of right-angle triangles. The side opposite the right angle is always the longest side, and is called the hypotenuse. Figure 5.1 Figure 5.1 shows a right-angle triangle. By using Pythagoras’ Theorem: where 2 2 AC (AB) (BC) adjacent opposite tan opposite sin hypotenus adjacent cosine hypotenus Theorem: AC2 = AB2 + BC2 B C A hypotenuse opposite adjacent
INTERMEDIATE MATHEMATICS (STEM C) 97C Example 1 Figure below shows a right-angle triangle, ABC. State the angle of BACcorrect to two decimal places. Solution: 1 24 sin 26 24 sin 26 67.38 A A Example 2 Find the perimeter of a right-angled tringle if the sides of the lengths is 10 cmand 15 cm respectively. Solution: 2 2 Hypotenuse 10 15 325 18.03 cm Perimeter 10 15 18.03 43.03 cm 24 cm 15 cm 10 cm 26 cm A B A B C
INTERMEDIATE MATHEMATICS (STEM C) 98Let's Master Mathematics! 1. Figure below shows a right-angle triangle PQR where PR 13 cm and QR 18 cm. Find : a) QPR . b) PQR . c) the length of PQ. d) the perimeter of the triangle PQR. 18 cm 13 cm R Q P
INTERMEDIATE MATHEMATICS (STEM C) 992. Find the value of x for the triangle given below: a) x 11 cm 9 cm 8 cm x 5 cm x cm4 cm B C17 cm 11 cm A 55.15
INTERMEDIATE MATHEMATICS (STEM C) 1005.2 Sine Rule and Cosine Rule Given any triangle with sides a, b, c and opposite angles A, B and C respectively, The Sine Rule is used to solve triangles when given: two sides with one angle opposite to one of the sides of the triangle or two angles and one side of the triangle. If given any triangle ABC with the sides a, b, c and opposite angles A, B and Crespectively, The Cosine Rule is used to solve the triangles when given: three sides of a triangle or two sides and one angle in between the sides. sin A sinB sinCa b c or a b c sin A sinB sinC A c a b B c a b 2ab cos Cb a c 2ac cosBa b c 2bc cos A2 2 2 2 2 2 2 2 2 A B C b c a C
INTERMEDIATE MATHEMATICS (STEM C) 101Example 1 (The Sine Rule) In ABC , A 55.3 , B 40.5 and c 18 cm. Solve the triangle, give your answers correct to two decimal places. Solution: 84.2 C 180 55.3 40.5 sin55.3 sin84.2 BC 18 BC 14.87cm sin 40.5 sin84.2 AC 18 AC 11.75cm Example 2 In ABC,B 25 , b 23cm and c 12.5cm. Find c and length of a. Solution: sin25 sinC 23 12.5 sin C 0.2297 C 13.28 sin141.72 sin25 a 23 a 33.72cm 141.72 A 180 25 13.28 40.5 55.3 C A B 18cm 25 C A B 12.5cm 23cm
INTERMEDIATE MATHEMATICS (STEM C) 102Example 3 (The Cosine Rule) Given ABC with a 6.7 cm b 8.12 cm and c 11.2 cm. Find A, B andC. Solution: A 36.36 cos A 0.8053 181.89cos A 146.48 44.89 101.37 181.89cos A (6.7) (8.12) (11.2) 2(8.12)(11.2)cos A a b c 2bc cos A 2 2 2 2 2 2 2 2 2 2 2 2 b a c 2bc cosB (8.12) (6.7) (11.2) 2(6.7)(11.2)cosB cosB 0.6956 B 45.92 97.72 C 180 36.36 45.92 Example 4 Given ABC with a 7 cm b 5.6 cm and C 112 . Solve the triangle. Solution: c 10.475cm 109.729 (7) (5.6) 2(7)(5.6)cos112 c a b 2abcosC 2 2 2 2 2 A 38.29 cosA 0.7849 7 (5.6) (10.475) 2(5.6)(10.475)cosA a b c 2bc cosA 2 2 2 2 2 2 29.71 B 180 38.29112
INTERMEDIATE MATHEMATICS (STEM C) 103Example 5 Find the length of QR and the angle of PQR if the length of r 5cm, q6.2cmand QPR 120 . Solution: p 9.72cm 94.44 (6.2) (5) 2(6.5)(5)cos120 p q r 2qr cosP 2 2 2 2 2 Let's Master Mathematics! 1. Given ABC where a 3cm,c 4.4cm and C 47 . Solve the triangle. 2. PQR is a triangle with QPR 89 ,PR 5.5cm and QR 18.6cm . Find a) PQR. b) the length of PQ.
INTERMEDIATE MATHEMATICS (STEM C) 1043. Given ABC where CBA 65 , CAB 51 and BC 7cm . Find the length of a) AB. b) AC. 4. Given MNKwith sides NK 70cm ,MK 92 cm and MKN 20 . Find a) the length of MN. b) MNK . 5. Given PQR as shown in the diagram below: FindPRM . M 42 108 18 cm 21.3c m R P Q 27 cm
INTERMEDIATE MATHEMATICS (STEM C) 1056. The sides of ABC are 4 cm, 5.6 cm and 8 cm. Find the angles of the triangle. 5.3 Area and Perimeter of Triangle The formula of the area of any triangle ABC is given by: ac sinB 2 1 ab sinC 2 1 bc sinA 2 1 Area A B C b c a
INTERMEDIATE MATHEMATICS (STEM C) 106Example 1 Find the area of a triangle if given that A 102 , b 22.5 cm and c 12 cm. Solution: 2 132.05cm (22.5)(12)sin102 2 1 bc sinA 2 1 A Example 2 Calculate the area of ABC if A 55.4 , AB 7.3 cm and AC 5.4 cm . Solution: 2 16.22cm (5.4)(7.3)sin55.4 2 1 bc sinA 2 1 A Example 3 Figure above shows two triangles of ADC and ABD. Given BD = 9 cm, AB = 8 cm, BC=13cm, CAD 29 and ADB 56 . Calculate a) the perimeter of triangle ABD. b) the area of ABC . 56 29 D C B A
INTERMEDIATE MATHEMATICS (STEM C) 107Solution: a) the perimeter of triangle ABD. 1 8 9 sin56 sin A 9sin56 sin A 8 0.9327 A sin 0.9327 68.85 length of AD: AD 8 sin55.15 sin56 AD 7.92 cm Perimeter = 8+9+7.92 = 24.92 cm b) the area of ABC 2 1 Area a c sinB 2 1 13 8 sin55.15 2 42.67cm B 180 68.85 56 55.15
INTERMEDIATE MATHEMATICS (STEM C) 108Let's Master Mathematics! 1. Given B 40.3 , AB 8.2cm and BC 4.1cm. Find the area of ABC . 2. Given the sides of a triangle are 5.5cm, 7 cm and 9.2 cm. Find the area of thetriangle. 3. Find the perimeter and the area of triangle below:
INTERMEDIATE MATHEMATICS (STEM C) 1094. Figure shows a triangle where OB 12cm,OC 2cm,OA 14cm and AOB 80 . Find a) the length of AB c) the perimeter of triangle ABC b) the area of ABC d) the angle of OAC 80 B A 14cm 12cm 2cm C
INTERMEDIATE MATHEMATICS (STEM C) 110EXERCISE 1. Figure below shows PQ 4cm , QR 5cm and QPR 30 . a) Find PRQ in degree. b) Calculate the length of PR. 2. Given that PQ = 11.5cm, PR = 8.1cm and QR = 13.7cm, find the largest angle usingcosine rule. 3. Figure shows a triangle where FD 4.2cm , D 102 and E 35. Find the lengthof ED. R Q P E F 102o 35o 4cm 5cm Q P R D
INTERMEDIATE MATHEMATICS (STEM C) 1114. Figure below shows two triangles of ABD and BCD. The area of triangle BCDis12cm2 and BCD is an acute angle. Calculate a) BCD . b) the length of BD (in cm). c) ABD . 5. Figure shows a triangle PQR with PQ 7cm , QR 4cmand PR9cm. CalculatePQR and the area of the triangle PQR. 6. Figure below shows triangle ABD and ADC. Given that BAC 82.5, AD7cm, AB 10cm and BC 12cm . Calculate a) ACB . b) the length of BD. c) the area of triangle AD 82.5 P Q S R 10cm 7cm A B D C 39 C D B A 6 cm 10 cm 7 cm
INTERMEDIATE MATHEMATICS (STEM C) 1127. Given that OPQ is a triangle where OP 20.5cm, OQ 31.6cmand POQ36. Find the area of triangle OPQ. 8. Figure below shows that C 56 , AC 18cmand BC 30cm. Find: a) the length of AB. b) ABC 9. PQR is a triangle where PQ 9cm , PR 18cmand QR 12cm. Find Qandthearea of the triangle. 10. PQR is a triangle where PQ 7cm , PR 5cmand Q 40 . Find Randthearea of the triangle. 11. Figure above shows a triangle ABC where AB 7cm, BC11.2cmandABC 87 . Find: a) the length of AC. b) BAC . A B 18cm 30cm C A B C
INTERMEDIATE MATHEMATICS (STEM C) 11312. Solve for x. 13. Given C 43 , AB 45mm and BC 38mm. Find A and the area of triangle ABC. 14. Figure above shows a triangle ABC where AC 4cm, CD 6.5cm, AD8cmandABC 45 . Find a) ACD. b) the area of triangle ACD. c) the length of AB. B 45 C A B x 15cm 22cm 13cm C A B A C D
INTERMEDIATE MATHEMATICS (STEM C) 114Answers 1. a) 126.42 b) 8.04cm 2. 86.87 3. 4.994cm 4. a) 34.85 b) 6.083cm c) 118.49 5. PQR 106.6 , Area =13.42cm2 6. a) 55.71 b) 9.60cm c) 8cm2 7. 190.39cm2 8. a) 24.9cm b) 36.82 9. 117.28 and A=48cm2 10. 64.15 and A=16.97cm2 11. a) 12.89cm b) 60.19 12. 103.34 13. A 35.16 , Area of triangle: 836.81mm2 14. a) 96.35o b) 12.92cm2 c) 5.62cm