Our last new Section…………5.6

Deriving the Law of Cosines a c b AB(c,0) C(x, y) a c b AB(c,0) C(x, y) a c b AB(c,0) C(x, y) In all three cases: Rewrite:

Deriving the Law of Cosines a c b AB(c,0) C(x, y) a c b AB(c,0) C(x, y) a c b AB(c,0) C(x, y) Set a equal to the distance from C to B using the distance formula:

Deriving the Law of Cosines

Law of Cosines Let ABC be any triangle with sides and angles labeled in the usual way. Then Note: While the Law of Sines was used to solve AAS and ASA cases, the Law of Cosines is required for SAS and SSS cases. Either method can be used in the SSA case, but remember that there might be 0, 1, or 2 triangles.

Guided Practice Solve ABC, given the following. A B C c

Guided Practice Solve ABC, given the following. A BC 7 9 5

Recall some diagrams : a c b AB(c,0) C(x, y) a c b AB(c,0) C(x, y) a c b AB(c,0) C(x, y) For all three triangles: Rewrite: This can be considered the height of each triangle, while side c would be the base…

Area of a Triangle Area = (base)(height) Generalizing: Note: These formulas work in SAS cases… 1 2 = (c)(b sinA) 1 2 = bc sinA 1 2 Area =

Theorem: Heron’s Formula Let a, b, and c be the sides of ABC, and let s denote Clearly, this theorem is used in the SSS case… the semiperimeter (a + b + c)/2. Then the area of ABC is given by: Area =

Guided Practice Find the area of the triangle described. An SAS case!!! (a) Area = An SAS case!!! (b) Area =

Guided Practice Decide whether a triangle can be formed with the given side lengths. If so, use Heron’s formula to find the area of the triangle. Yes  the sum of any two sides is greater than the third side!!! (a) Can a triangle be formed? Find the semiperimeter: Heron’s formula:

Guided Practice Decide whether a triangle can be formed with the given side lengths. If so, use Heron’s formula to find the area of the triangle. No  b + c < a (b) Can a triangle be formed?

Guided Practice Decide whether a triangle can be formed with the given side lengths. If so, use Heron’s formula to find the area of the triangle. Yes  the sum of any two sides is greater than the third side!!! (c) Can a triangle be formed? Find the semiperimeter: Heron’s formula:

Guided Practice p.494: #36 (a) Find the distance from the pitcher’s rubber to the far corner of second base. How does this distance compare with the distance from the pitcher’s rubber to first base? A (First base) Second base (Pitcher’s rubber) B (Home plate) C 90 ft 60.5 ft 45 c The home-to-second segment is the hypotenuse of a right triangle, which has a length of… Distance from pitcher to second:

Guided Practice p.494: #36 (a) Find the distance from the pitcher’s rubber to the far corner of second base. How does this distance compare with the distance from the pitcher’s rubber to first base? A (First base) Second base (Pitcher’s rubber) B (Home plate) C 90 ft 60.5 ft 45 c Solve for c with the Law of Cosines: ft

Guided Practice p.494: #36 (b) Find angle B in triangle ABC. A (First base) Second base (Pitcher’s rubber) B (Home plate) C 90 ft 60.5 ft 45 c Again, the Law of Cosines: ft

Whiteboard Practice Solve ABC, given the following. C BA 12 c 11 51

Whiteboard Practice Solve ABC, given the following. C BA 12 c 11 51

Whiteboard Practice Solve ABC, given the following. C BA 8.9 c  No real solutions!!!  No triangle is formed!!!