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Presentation transcript:

The Law of Sines

Quick Review

Quick Review Solutions

What you’ll learn about Deriving the Law of Sines Solving Triangles (AAS, ASA) The Ambiguous Case (SSA) Applications … and why The Law of Sines is a powerful extension of the triangle congruence theorems of Euclidean geometry.

Law of Sines

Example Solving a Triangle Given Two Angles and a Side

Example Solving a Triangle Given Two Sides and an Angle (The Ambiguous Case)

Example Finding the Height of a Pole x 15ft 15 º 65 º B A C

Example Finding the Height of a Pole x 15ft 15 º 65 º B A C

The Law of Cosines

Quick Review

Quick Review Solutions

What you’ll learn about Deriving the Law of Cosines Solving Triangles (SAS, SSS) Triangle Area and Heron’s Formula Applications … and why The Law of Cosines is an important extension of the Pythagorean theorem, with many applications.

Law of Cosines

Example Solving a Triangle (SAS)

Area of a Triangle

Heron’s Formula

Example Using Heron’s Formula Find the area of a triangle with sides 10, 12, 14.

Example Using Heron’s Formula Find the area of a triangle with sides 10, 12, 14.

Chapter Test

9. A hot-air balloon is seen over Tucson, Arizona, simultaneously by two observers at points A and B that are 1.75 mi apart on level ground and in line with the balloon. The angles of elevation are as shown here. How high above ground is the balloon? 10. A wheel of cheese in the shape of a right circular cylinder is 18 cm in diameter and 5 cm thick. If a wedge of cheese with a central angle of 15 º is cut from the wheel, find the volume of the cheese wedge.

Chapter Test Solutions

9. A hot-air balloon is seen over Tucson, Arizona, simultaneously by two observers at points A and B that are 1.75 mi apart on level ground and in line with the balloon. The angles of elevation are as shown here. How high above ground is the balloon? 10. A wheel of cheese in the shape of a right circular cylinder is 18 cm in diameter and 5 cm thick. If a wedge of cheese with a central angle of 15 º is cut from the wheel, find the volume of the cheese wedge. ≈0.6 mi 405π/24≈53.01