1. 9.11a The Area of a Triangle. The Law of Sines
Rita Korsunsky

2. The area K of triangle ABC is given by each of the following formulas:
Area Formulas  b c h   a
The area K of triangle ABC is given by each of the following formulas:

3. Example 1 Q P R

4. Example 2 20 20

5. Find the area of triangle ABC if
Example 3
Find the area of triangle ABC if
a = 35, c = 42, and  = 115°
c = 42
a = 35   
Solution
Example 4
A surveyor finds that the edges of a triangular lot measure 42.5m, 37.0m, and 28.5m. What is the area of the lot? 37
42.5
28.5
Solution: Use Heron’s formula:
Therefore K =

6. LAW OF SINES:
Proof a b c h C B A H

7. where a, b, and A are given and A is acute
SSA:
where a, b, and A are given and A is acute A b a C c B A b a C c
No Solution
One Solution C b a A c B A b C c
One Solution
Two Solutions

8. Example 1: SSA 1. a = 2, b = 4, A = 22° 2. B = 40°, a = 12, b = 6 A C
Determine whether there are 0, 1, or 2 triangles possible for each of the following sets of measurements.
1. a = 2, b = 4, A = 22°
2. B = 40°, a = 12, b = 6 A C c C B c
40° B 6 12
22° A 4 2
1.49 < 2 < 4
Two Solutions
No Solution

9. Example 2: A 15 11 C C = 34.178° B A = 15.822° a = 5.339 Given:
130°
34.178° C
C = ° B
Solve the triangle
Given:
B = 130°
c = 11
b = 15
A = °
a = 5.339

10. Example 4: Ship Q P 66º 44º 16 km S q p 70º = ?
From lighthouses P and Q, 16 km apart, a disabled ship S is sighted. If SPQ = 44º and SQP = 66º, find the distance from S to the nearest lighthouse.
Ship Q P
66º
44º
16 km S q p
S = 180º - 66º - 44º = 70º
70º
p = km
= ?
The ship is km from the nearest lighthouse, Q

11. Example 5: C B
a=10 K
b=8 A

12. C A B
b=8
a=10 C A B
b=8
a=10
(1)
(2)