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Geometry 7-6 Circles, Arcs, Circumference and Arc Length
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Review
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Areas
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Area Area of a Triangle
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Theorem The Pythagorean theorem In a right triangle, the sum of the squares of the legs of the triangle equals the square of the hypotenuse of the triangle A C b B a c
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Theorem Converse of the Pythagorean theorem If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. A C b B a c
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Converse of Pythagorean
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Theorem 45° – 45° – 90° Triangle In a 45° – 45° – 90° triangle the hypotenuse is the square root of two times as long as each leg
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Theorem 30° – 60° – 90° Triangle In a 30° – 60° – 90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is the square root of three times as long as the shorter leg
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Area
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Vocabulary
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Area
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Definitions - Review
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Circle Vocabulary (review) Circle – The set of all points in a plane that are equidistant from a given point Center – Equidistant point of a circle Radius – Distance from the center of a circle to a point on the circle Diameter – Distance from a point on the circle to another point on the circle through the center of the circle Congruent Circles – Circles with congruent radii Central Angle – Angle with vertex at the center of the circle
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Vocabluary New Definitions
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Arc Vocabulary Part of the circle that measures between 180° and 360° Major Arc
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Arc Vocabulary Part of the circle that measures between 0° and 180° Minor Arc
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Arc Vocabulary An arc whose endpoints are the endpoints of a diameter of a circle Semicircle
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Arc Vocabulary The measure of the arcs central angle Measure of a Minor Arc The difference between 360° and the measure of its associated minor arc Measure of a Major Arc
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Arc Vocabulary Intercepted Arc Arc within an angle
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Arc Vocabulary (Summary) An angle whose vertex is the center of a circle Central Angle Part of the circle that measures between 180° and 360° Major Arc Minor Arc Part of the circle that measures between 0° and 180° Semicircle An arc whose endpoints are the endpoints of a diameter of a circle Measure of a Minor Arc Measure of a Major Arc The measure of the arcs central angle The difference between 360° and the measure of its associated minor arc
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New Theorem
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Example
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Three letters required for major arcs
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Example
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Arc Length What fraction of a circle is the arc? 1/4=90°/360°
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Arc Length What fraction of a circle is the arc? 1/2=180°/360°
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Arc Length What fraction of a circle is the arc? 1/3=120°/360°
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Arc Length The measure of an arc is calculated in units of degrees, but arc length is calculated in units of distance
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Investigation For circles T, O and P, calculate the following in your notes.
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Investigation Find the circumference for each circle.
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Investigation Find the Arc Length for each circle.
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Investigation What is the formula for arc length of a circle?
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Arc Length Theorem The length of an arc equals the fraction of the arc to the circle times the circumference Arc Length = Fraction * Circumference
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Example
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Arc Length - Example
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Practice
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Arc Length - Practice
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Practice
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Minor Arc
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Semicircle
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Major Arc
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Minor Arc
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Semicircle
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Minor Arc
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70°
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110°
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180°
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210°
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70°
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290°
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280°
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130°
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165°
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60°
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Arc Length - Practice
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Practice
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Homework Pages 390 – 393 1 – 8, 16 – 24 even, 28 – 38 even, 42, 44, 52, 63, 64, 68, 76 - 78
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