G-C: Circles

Circles

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Circles covers properties of circles including inscribed angles, central angles, arcs, chords, and tangent lines. Similarity of all circles is established, and arc length and sector area are derived from proportionality. Equation-based work on circles connects to coordinate geometry at the precalculus level.

G-C.1Prove that all circles are similar.G-C.2Identify 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.3Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.G-C.4(+) Construct a tangent line from a point outside a given circle to the circle.G-C.5Use and apply the concepts of arc length and areas of sectors of circles. Determine or 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.
Example Problems
The diameter of a circle is units.
What is the
radius of the circle?
A circle is centered at and has a radius of .
Where does the point
lie: inside the circle, on the circle, or outside the circle?
Convert the angle to radians.
Express your answer exactly.
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