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Lesson 10-R Chapter 10 Review
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Objectives Review Chapter 10 material
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Parts of Circles Circumference (Perimeter) Chord Radius Diameter
once around the outside of the circle; Formulas: C = 2πr = dπ Chord segment with endpoints of the edge of the circle Radius segment with one endpoint at the center and one at the edge Diameter segment with endpoints on the edge and passes thru the center longest chord in a circle is twice the length of a radius Other parts Center: is also the name of the circle Secant: chord that extends beyond the edges of the circle Tangent: a line (segment) that touches the circle at only one point
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Arcs in Circles Arc is the edge of the circle between two points
An arc’s measure = measure of its central angle All arcs (and central angles) have to sum to 360° If two arcs have the same measure then the chords that form those arcs have the same measure If a radius is perpendicular to a chord then it bisects the chord and the arc formed by the chord (example arc AED below) Major Arc (example: arc DAB) measures more than 180° more than ½ way around the circle Minor Arc (example: arc AED) measures less than 180° less than ½ way around the circle Semi-circle (example: arc EAB) measures 180° defined by a diameter BE is a diameter and AB = AD 120° B A C 120° 60° E D 60°
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Angles Associated with Circles
Name Vertex Location Sides Formula Example Central Center radii = measure of the arc BCD = 110° Inscribed Edge chords = ½ measure of the arc BAD = 55° Interior Inside = average of the vertical arcs EVH = 73° Exterior Outside Secants / Tangents = ½ (Big Arc – Little Arc) = ½ (Far Arc – Near Arc) NVM = 30° minor arc LK = 10° minor arc NM = 70° E G F C 110° H 36° V minor arc FG = 110° minor arc EH = 36° V K L M N A D B C 110° minor arc BD = 110° 10° C 70°
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Segments Inside/Outside of Circles
Segments that intersect inside or outside the circle have the length of their parts defined by: Two Chords Inside a Circle Two Secants From Outside Point Secant & Tangent From Outside Point J J K 4 5 3 6 L K K 3 L 4 J T 8 9 7 6 11 N N M M M LJ · JM = NJ · JK 3 8 = 6 4 JL · JN = JK · JM 5 12 = 4 15 JT · JT = JK · JM 6 6 = 3 12 Inside the circle, it’s the parts of the chords multiplied together Outside the circle, it’s the outside part multiplied by the whole length OW = OW
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Tangents and Circles Tangents and radii always form a right angle
We can use the converse of the Pythagorean theorem to check if a segment is tangent The distance from a point outside the circle along its two tangents to the circle is always the same distance S T J C Example 1 Given: JT is tangent to circle C JC = 25 and JT = 20 Find the radius Example 2 Given: same radius as example 1 JC = 25 and JS = 16 Is JS tangent to circle C? JC² = JT² + TC² 25² = 20² + r² 625 = r² 225 = r² 15 = r JC² = JS² + SC² 25² = 16² + 15² 625 = 625 ≠ 481 JS is not tangent
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Equation of Circles A circle’s algebraic equation is defined by: (x – h)² + (y – k)² = r² where the point (h, k) is the location of the center of the circle and r is the radius of the circle Circles are all points that are equidistant (that is the distance of the radius) from a central point (the center) Midpoint of the endpoints of diameter is the center Distance formula between the center and a point on the circle can find the radius
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Summary & Homework Summary: Homework:
All my life’s a circle -- Harry Chapin Homework: study for the test
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