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

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:

Example 1 Q P R

Example 2 20 20

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 =

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

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

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

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

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 = 11.829 km = ? The ship is 11.829 km from the nearest lighthouse, Q

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

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