In section 9.2 we mentioned that by the SAS condition for congruence, a triangle is uniquely determined if the lengths of two sides and the measure of the included angle are known.

By the side-side-side (SSS) condition for congruence, a triangle is also uniquely determined if the length of the sides are known.

The law of cosines can be used to solve a triangle in either of these two cases.

The Law of Cosines2 2 2 2 cosa b c ab A 2 2 2 2 cosb a c ac B 2 2 2 2 cosc a b ab C

2 2 2When C = 90 the law of cosines reduces to .c a b 2 2 2When C is acute, is less than a by the amount 2 cos .c b ab C

2 2 2When C is obtuse, cos is negative and so is greater than .C c a b

Example 1. Suppose that two sides of a triangle have lengths 3 cm and 7 cm and that the angle between them measures 130º. Find the length of the third side.

Make a sketch 3cm

7cm130

c

2 2 23 7 2 3 7 cos130c 85 85 9.22 cmc

If we solve the law of cosines for Cos C, we obtain2 2 2

2 2 2

2 2 2

2 2 2

2 cos

2 cos

2 cos

cos2

c a b ab C

c ab C a b

ab C a b c

a b cC

ab

Example 2. The lengths of the sides of a triangle are 5, 10, and 12. Solve the triangle.

Make a sketch

5

10 12

2 2 25 10 12cos .19

2 5 10

101.0

2 2 212 10 5cos 0.9125

2 12 10

24.1

180 101.0 24.1 54.9

Using the law of cosines, we can easily identify acute and obtuse angles.

The law of sines does not distinguish between obtuse and acute angles, however, because both types of angle have positive sine values.

Example 3. In the diagram at the right, AB = 5, BD = 2,

DC = 4, and CA = 7. Find AD.

Remember SSA is the ambiguous case