Arc lengths, sectors, and radians
• Use similarity to derive the fact that length of an intercepted arc of an angle is proportional to the length of the radius of the circle. • Define radian measure of an angle. • Derive a formula for the area of a sector. • Use the formula in solving problems.
Students begin this unit by working with circles on the coordinate plane and use the Pythagorean Theorem to derive the equation of a circle. They learn about radian measure and use similarity to derive the fact that the length of the arc intercepted by an angle is proportional to the radius of the circle. They develop a formula for the area of a sector of a circle and examine properties of central and inscribed angles in a circle along with their subtended arcs. Students also examine properties of tangent lines, radii, and intersecting chords and apply these properties to solve problems in a variety of contexts, including real-world situations.