Coordinate+Geometry+with+Circles

SMCCII Unit Goal: Use coordinates to prove simple geometric theorems involving circles algebraically.
 * 1) G.C.2 –Identify and describe relationships among inscribed angles, radii, and chords.
 * 2) 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.
 * 3) G.GPE.4-Use coordinates to prove simple geometric theorems algebraically.
 * 4) G.GPE.6-Find the point on a directed line segment between two given points that partitions the segment in a give ratio (midpoint).

Surface ideas such as. . . Students will need the Right Angle Task worksheet and string if desired Launch: Introduce the task. Let the students read the directions for #1 first and complete the task. Explore: While students are creating their circles, verify students are creating circles rather than ellipses. Let the groups work on the problem if a group gets done early, ask them to find another right angle. Discuss: Did anyone find that the inscribed angle was a right angle? How did you know it was a right angle? Students will need the Circle Equation worksheet. Launch: Introduce the task. Explore: Discuss: Surface ideas such as. ..
 * //__Develop Understanding Task__//**
 * Review distance formula from CCSI (G.GPE.7)
 * Review slope formula from CCSI (G.GPE.5)
 * Inscribed angles on a diameter are right angles (G.C.2)
 * Properties of a rectangle
 * Task #0:** __Right Angle Task__ (It is suggested that students complete the first problem, Kaleb’s shed, in a complete Teaching Cycle, before working on #2-5.)
 * Task #1:** Circle Equation Task (Consider bringing students back together after one or two components to maintain progress.)
 * If you selected another point on the circle would the relationship stay the same?
 * Students may use Pythagorean Theorem or the distance formula to find the distance.
 * Using geometry software, illustrate that moving Point A maintains the same relationship with the radius and does not change the equation of the circle.

Task #3: Inscribed Angles Task Task #4: Inscribed Isosceles Trapezoid Task Task #5: Inscribed Isosceles Triangle Task Task #6: Circle Area Task
 * //__Solidify Understanding Lesson__//**
 * //__Practice Understanding Lesson__//**