1. HKDSE MATHEMATICS Ronald Hui Tak Sun Secondary School

2. MISSING HOMEWORK  RE2  Kelvin  SHW2-R1  Kelvin, Charles  SHW2-P1  Daniel, Kelvin, Sam L, Charles 22 October 2015 Ronald HUI

3. MISSING HOMEWORK  SHW3-01  Kelvin, Charles  SHW3-A1  Charles  SHW3-B1  Charles  SHW3-C1  Daniel, Kelvin(RD), Sam L, Charles 22 October 2015 Ronald HUI

4. MISSING HOMEWORK  SHW3-E1  Kelvin, Charles  SHW3-R1  Daniel, Charles  SHW3-P1  Kelvin, Charles, Isaac, Macro S (RD) 22 October 2015 Ronald HUI

5. MISSING HOMEWORK  Chapter 4 HW  SHW4-01  SHW4-A1  SHW4-B1  SHW4-C1  SHW4-D1  SHW4-R1  SHW4-P1  Deadline: 26 Nov (Thursday) 22 October 2015 Ronald HUI

6. Book 5A Chapter 5 Area of a Triangle

7. In trigonometry, we usually represent the angles and sides of a triangle, say △ ABC, as follows: (a) The interior angles are denoted by capital letters A, B and C respectively. A B C a b c (b) The lengths of the sides opposite to angles A, B and C are denoted by the corresponding small letters a, b and c respectively.

8. In trigonometry, we usually represent the angles and sides of a triangle, say △ ABC, as follows: A 5 cm B 7 cm C 62  A = a = b = Example: 7 cm5 cm 62  A B C a b c

9. There is a formula to calculate the area of a triangle when only two of its sides and their included angle are known. For example: 9 cm 7 cm 40  7 cm 9 cm 140 

10. Suppose only two sides a and b of △ ABC and their included angle C are given. Consider the following two cases. h D A B C b a Case 1: C is an acute angle. Consider △ ADC. h D A B C b a Case 2: C is an obtuse angle. Consider △ ADC. ◄  ACD = 180   C ◄ sin (180   C) = sin C To find the area of △ ABC, we have to find its height first. Given two sides and their included angle

11. Case 1: C is an acute angle. h D A B C b a Case 2: C is an obtuse angle. In both cases, the height of the triangle with base a is equal to b sin C. h D A B C b a

12. Case 1: C is an acute angle. h D A B C b a Case 2: C is an obtuse angle. Thus, we have: Area of △ ABC h D A B C b a

13. Note: The above formula is also valid for right-angled triangles. When C = 90 , 2 1 ab sin C = = bc sin A 2 1 = ac sin B 2 1 the included angle of a and b. Similarly, for any △ ABC, we have: Area of △ ABC area of △ ABC the included angle of b and c. the included angle of a and c. b c A B a C b a C b c A c a B

14. Area of △ ABC d. p.) 1 to (cor. Find the area of △ ABC correct to 1 decimal place. C 105  A B 6 cm 10 cm b = 6 cm, c = 10 cm, A = 105 

15. Follow-up question The figure shows △ ABC with area 10 cm 2. If AC = 7 cm and  BAC = 75 , find the value of y correct to 3 significant figures. Area of △ ABC A B C y cm 7 cm area = 10 cm 2 75  (cor. to 3 sig. fig.)

16. When only the lengths of the three sides of a triangle are known, could we find the area of the triangle? Yes, we can use Heron’s formula to find the area of the triangle.

17. Area of △ ABC where s s is called the semi-perimeter. Heron’s formula s(s  a)(s  b)(s  c), A C a b c B

18. A B C 5 cm 7 cm 8 cm Area of △ ABC Let me show you how to find the area of △ ABC correct to 3 significant figures. Let. 2 cba s   cm 10  ∴ cm 2 578   s ))( (csbsass  2 cm (10 – 5)(10 – 7)(10 – 8)10  fig.) sig. 3 to (cor. cm 3.17 2 

19. Follow-up question There figure shows △ ABC, where AB = 9 cm, BC = 12 cm and AC = 13 cm. (a) Find the area of △ ABC. (b) Find the length of BD. (Give your answer correct to 3 significant figures.) Area of △ ABC (a) Let ∴ A B C 9 cm 13 cm 12 cm D (cor. to 3 sig. fig.)

20. (b) ∵ Area of △ ABC ∴ (cor. to 3 sig. fig.) Follow-up question There figure shows △ ABC, where AB = 9 cm, BC = 12 cm and AC = 13 cm. (a) Find the area of △ ABC. (b) Find the length of BD. (Give your answer correct to 3 significant figures.) A B C 9 cm 13 cm 12 cm D