Warm Up1) Find the height of the triangle below:

2) Find the area of the triangle below:

A

B

C

10

15

8

20º

32130.8°

9.2: Area of a Triangle

9.2 Area of a Triangle

Objectives:1. Find the area of a non-right

triangle.

Vocabulary: base, altitude, area, parallelogram, quadrilateral

The area A of a triangle equals one-half the product of two of its sides times the sine of its included angle.

A = ½ a•b•sin(C)

A = ½ a•c•sin(B)

A = ½ b•c•sin(A)

Find the area of the quadrilateral.

A

B

C

8

5

10

D130º

9.4: Law of Cosines

8

Deriving the Law of Cosines

Write an equation using Pythagorean theorem for shaded triangle.

b h a

k c - kA B

C

c

sincos

h b Ak b A

2 22

2 2 2 2 2 2

2 2 2 2 2

2 2 2

sin cos

sin 2 cos cos

sin cos 2 cos

2 cos

a b A c b A

a b A c c b A b A

a b A A c c b A

a b c c b A

Note:

capital letters always stand for __________!

lower-case letters always stand for ________!

When to use the Law of Cosines: SSS

SASangles

sides

A

B

C

a

b

c

Cabbac

Baccab

Abccba

cos2

cos2

cos2

222

222

222

The Law of Cosines!

Let’s do some problems! Ex. 1: Use the Law of Cosines to solve each triangle. Round any decimal

answers to the nearest tenth.

a. .19and,16,35,In cbAmABC

16

35˚

B

C

A

19

Abccba cos2222 a 35cos191621916 222a

...95.1182 a

9.10aBaccab cos2222

Bcos199.102199.1016 222

Bcos2.41481.223

B3.57

Bcos2.41481.479256 Bcos2.41481.223

Bcos...5403. 7.87C

57.3˚

b. .16and,20,8,In derRED ER

D

820

16Redder cos2222

Rcos1620216208 222

Rcos640592

R3.22

Rcos64065664 Rcos640592

Rcos925.

Erddre cos2222 Ecos168216820 222

Ecos25680

E2.108

Ecos256320400 Ecos25680

Ecos3125. 5.49D

22.3 1̊08.2˚

.40and,20,16,In CmtaCAT T

A

C

1620

c

40˚

53.0˚

Cattac cos2222 40cos201622016 222c

...73.1652 c

9.12cActtca cos2222

Acos209.122209.1216 222

Acos516

41.310

A0.53

Acos51641.566256 Acos51641.310

Acos...6015. 0.87T

Now you try!

Exit Slip

Solve the triangle:p = 3, q = 8, <R = 50˚