M3U8D1 Warm Up x 120 4x Simplify each expression:

M3U8D1 REVIEW: Vocabulary of Circles and Area of Sectors in radians and degrees AND Practice with Relationships of Circles, Angles, Arcs, and Sectors OBJ: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. G-C.5 AND Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. G-C.2

TermDefinitionPicture Circle All points equidistant from a given point called the center Chord A segment whose endpoints are on the circle Radius A segment from the center to a point of the circle Diameter A special chord that passes through the center Secant A line which intersects the circle in two points Tangent A line (in the same plane) which intersects the circle in one and only one point Arc An arc is a portion of the circumference of a circle. Minor arc Arc that measures less than 180˚ Major Arc Arc that measures greater than 180˚ Semicircle Arc of a circle that measures 180 ⁰ Distribute Vocabulary List

What do you notice about the radius in each picture? The radius of a circle is perpendicular to the point of tangency.

Picture 1Picture 2Picture 3 Where is vertex? Center of circleOn the circle Outside of the circle Name of Angle Central AngleInscribed Angle Circumscribed Angle Therefore (Formula)

180˚ ½ 18  180˚ 360 or 18  36  1/2 90˚ 360 1/4 99 90˚ 360 or 9  36  1/4 120˚ 360 1/3 12  120˚ 360 or 12  36  1/3 x˚ 360 x 360 or Column C and Column F are the same

How can we use ratio and proportions to help us find the area of a sector? Answers will vary. The idea is for the students to come up with the “formula” and/or the “proportion” on their own. The idea is for the students to think of Area of a Sector as a portion/fraction/proportion of the total Area of the circle.

You Try… Column C and Column F are the same

How can we use ratio and proportions to help us find the area of a sector? As stated above, the class should benefit from the student ’ s presentations of their tables in hopes that some groups/students used the “ formula ” and others used the “ proportion ”. *You can connect this to “ half past ” and “ quarter after ” / ” quarter til ” on a clock also.

Classwork M3U8D1 Vocabulary Review M3U8D1 Modeling with Trig Functions #1-7 Homework M3U8D1 Modeling with Trig Functions #8