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Chapter 9 The Area of a Triangle

Case 1 Two angles and one side are known Case

2 Two sides and one angle opposite one of those

sides are known Case 3 Two sides and the

included angle are known Case 4 All three sides

are known

C

b

a

B

A

c

Chapter 9 The Area of a Triangle

- Formulas exist for finding the area of a triangle

for - Case 1 Two angles and one side are known
- Case 3 Two sides and the included angle are

known - Case 4 All three sides are known
- Start with Case 3
- Draw line h
- Assume we
- know A, b and c

C

b

a

h

B

A

c

Chapter 9 The Area of a Triangle

Assume we know A, b and c K Area K

½ch

C

b

a

h

B

A

c

Chapter 9 The Area of a Triangle

Assume we know A, b and c K Area K

½ch But sinA h/b So h bSinA K

½cbsinA

C

b

a

h

B

A

c

Chapter 9 The Area of a Triangle

K ½bcsinA K ½acsinB K ½absinC

C

b

a

B

A

c

Chapter 9 Example

Example of Case 3 - Find the area of this

triangle A 33º b 40 c 45

C

b

a

B

A

c

Chapter 9 Example

A 33º b 40 c 45 K ½bcsinA

½(40)(45)sin33 490.2

C

b

a

B

A

c

Chapter 9 The Area of a Triangle

Case 1 Two angles and one side are known Case

3 Two sides and the included angle are

known Case 4 All three sides are known For

Case 1 K ½bcsinA But replace c by c

C

b

a

bsinC sinB

B

A

c

Chapter 9 The Area of a Triangle

K K K

a2sinB sin C 2 sin A

b2sinA sin C 2 sin B

C

c2sinA sin B 2 sin C

b

a

B

A

c

Chapter 9 Example

Example of Case 1 A 82º B 59º b 12

C

b

a

B

A

c

Chapter 9 Example

Example of Case 1 A 82º B 59º b 12 K

52.35

C

b

a

b2sinA sin C 2 sin B

B

A

c

Chapter 9 Example

Case 4 Use Herons Formula K v s(s

a)(s b)(s c) Where s ½(a b c) s is

the semiperimeter of the triangle

C

b

a

B

A

c

Homeworkp. 571

- 2, 4, 6, 8, 10, 12, 26

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