G-C. Circles

G-C.A. Understand and apply theorems about circles

• Two Wheels and a Belt

G-C.A.1. Prove that all circles are similar.

• Similar circles

G-C.A.2. 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.

• Neglecting the Curvature of the Earth
• Right triangles inscribed in circles I
• Right triangles inscribed in circles II
• Tangent Lines and the Radius of a Circle

G-C.A.3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

• Circumcenter of a triangle
• Circumscribed Triangles
• Inscribing a circle in a triangle I
• Inscribing a circle in a triangle II
• Inscribing a triangle in a circle
• Locating Warehouse
• Opposite Angles in a Cyclic Quadrilateral
• Placing a Fire Hydrant

G-C.A.4. Construct a tangent line from a point outside a given circle to the circle.

• Tangent to a circle from a point

G-C.B. Find arc lengths and areas of sectors of circles

• Orbiting Satellite
• Setting Up Sprinklers
• Two Wheels and a Belt

G-C.B.5. 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.

• Mutually Tangent Circles