6.+Circles+With+and+Without+Coordinates

Standards
G.C.1. Prove that all circles are similar. G.C.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 G.C.3. Construct 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.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.

G.GPE.1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. G.GPE.2. Derive the equation of a parabola given a focus and directrix. G.GPE.4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1,√3) lies on the circle centered at the origin and containing the point (0,2).

G.GMD.1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. G.GMD.3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Examples of Learning Cycled and Tasks Learning Cycle G.GPE.1 Inscribed.Central.Angles Equation of a Circle in Standard Form G.GPE.1 Murray Core Academy June 11-14, 2012 Inscribed and Circumscribed Circles __Salem Hills CA__ Find Arc Length of Circles, Copper Hills, Core Academy <span style="background-color: white; color: #333333; display: block; font-family: inherit; font-size: medium; text-align: left;">Equations of Circles-Pythagorean Theorem, Richfield CA G.GPE.1 <span style="background-color: white; color: #333333; display: block; font-family: inherit; font-size: medium; text-align: left;">Equations of Circles using sprinklers, Richfield CA G.GPE.1 <span style="background-color: white; color: #333333; display: block; font-family: inherit; font-size: medium; text-align: left;">Coordinate Geometry with Circles G.GPE.4., Cedar City