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Glenbard District 87 Course Title: Geometry Unit: 5 Unit Title: Circles With and Without Coordinates Stage 1 – Desired Results Established Goal(s): What relevant goals (e.g. Content standards, course This is chapter 12. or program objectives, learning outcomes, etc.) will this address? ● Understand and apply theorems about circles. ● Find arc lengths and areas of sectors of circles. ● Translate between the geometric description and the equation for a conic section. ● Use coordinates to prove simple geometric theorem algebraically. ● Apply geometric concepts in modeling situations. Understanding(s): Students will understand that… ● Circles can be defined by algebraic equations ● There is a relationship among angle measures created by chords, secants and tangents, arc measures and sector areas ● All circles are similar Knowledge: Students will know… 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. Essential Question(s): What provocative questions will foster inquiry, understanding, and transfer of learning? ● Why are circles important geometric figures? ● Where does pi come from? ● Why do arcs need to be measured in 2 different ways (measure vs. length)? Skills: Students will be able to … G.C.1 Prove that all circles are similar. 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 Vocabulary: Chord Secant Tangent Central Angle Inscribed Angle Circumscribed Angle (Tangent-tangent angle) Arc (Major & Minor) Sector Arc Length Arc Measure Radian Pi 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.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.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).*