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Topic VII Properties of Circles Opening routine Circle A has the center at the origin and a point D located on the circle at (1, 0). Circle B has a center at (1, 2) and the point E located on the circle at (4, 2). Logan performed two transformations on circle A to show that circle A is similar to circle B. He first dilates the circle with the center of dilation at the origin, and then translates the new circle. What are the algebraic descriptions of the two transformations? Topic VII Properties of Circles Opening routine The general rule for dilation by scale factor k with center of dilation at (a, b) is: Dk (x, y) = (a + k(x a), b + k(y b)) When the center of dilation is the origin, a = 0 and b = 0, so the rule simplifies to: Dk (x, y) = (kx, ky) As the radius of the circle increases from 1 to 3, the scale factor is 3, then the function for the dilation is: D3 (x, y) = (3x, 3y). The second transformation is a translation. The function for translations is: T(x, y) = (x + a, y + b) This translation slides point D’ in (3, 0) to point E in (4, -2), 1 unit to the right and 2 units down. Then the function for translation is: T(x, y) = (x + 1, y 2) Topic VII Properties of Circles Topic VII Properties of Circles Topic VII: Properties of Circles Topic VII Properties of Circles Objective: Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles. Essential Question: How can you determine the measures of central and inscribed angles of a circle? Topic VII Properties of Circles Vocabulary Circle - Is the set of all points in a plane equidistant from a given point called the center of the circle. Radius: Any segment with endpoints that are the center of the circle and a point on the circle. Chord: Segments with endpoints that are on the circle. Diameter: A chord that passes by the center of the circle. Arc: Is a part of the circle that is defined by two endpoints. Topic VII Properties of Circles Vocabulary Minor arc: Is the shorter arc connecting two endpoints. Major arc: Is the longest arc connecting two endpoints. Semicircle: Is an arc that measures 180o Central angle: An angle that intersects the circle in two points and has its vertex at the center of the circle. Inscribed angle: Is an angle that has its vertex on the circle and sides that contain chords of the circle. Topic VII Properties of Circles Topic VII Properties of Circles Topic VII Properties of Circles Topic VII Properties of Circles Topic VII Properties of Circles Topic VII Properties of Circles Topic VII Properties of Circles Topic VII Properties of Circles Topic VII Properties of Circles Topic VII Properties of Circles Topic VII Properties of Circles Topic VII Properties of Circles Topic VII Properties of Circles Topic VII Properties of Circles Properties of Circles Guided Practice – WE DO Topic VII Properties of Circles Topic VII Properties of Circles Topic VII Properties of Circles Guided Practice – WE DO If a square with sides of 14 cm is inscribed in a circle, what is the diameter of the circle? Topic VII Properties of Circles Guided Practice – WE DO As the triangle form by the diagonal and two of the sides of the square is a 45o - 45o - 90o, the length of the diagonal is 2 times the length of any of the sides. d = 14 2 Topic VII Properties of Circles Independent Practice – YOU DO Central and Inscribed Angles Worksheet Exercises 1 - 26 Topic VII Properties of Circles Re-teach MAFS.912.G-SRT.2.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Math Nation Section 7 Topic 1 Independent Practice Topic VII Properties of Circles Re-teach MAFS.912.G-CO.3.9: Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Math Nation Section 3 Topic 6 Independent Practice Topic VII Properties of Circles Re-teach MAFS.912.G-CO.3.10: Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Math Nation Section 8 Topic 4 Independent Practice Topic VII Properties of Circles Closure Essential Question: How can you determine the measures of central and inscribed angles of a circle? Topic VII Properties of Circles Homework Dilations Worksheet Due Monday February 6, 2017