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Analyzing Circles

Analyzing Circles. OBJECTIVES: Degree & linear measure of arcs Measures of angles in circles Properties of chords, tangents, & secants Equations of circles. About Circles. Definition : set of coplanar points equidistant from a given point P(center)  written P

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Analyzing Circles

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  1. Analyzing Circles OBJECTIVES: Degree & linear measure of arcs Measures of angles in circles Properties of chords, tangents, & secants Equations of circles

  2. About Circles • Definition: set of coplanar points equidistant from a given point P(center)  written P • Chord: any segment having endpoints on the circle • Radius(r): a segment from a point on the circle to the center • Diameter(d): chord containing the center of the circle • Circumference: the distance around the circle Circumference: C = πd = 2πr • Concentric circlesshare the same center & have different radius lengths

  3. Central angles have the vertex at the center of the circle The sum of non-overlapping central angles = 360° A central angle splits the circle into 2 arcs: minor arc: m major arc: m Adjacent arcs share only the same radius The measure of 2 adjacent arcs can be added to form one bigger arc. Arc Lengthis the proportion of the circumference formed by the central angle : Angles and Arcs Measure T V. P L

  4.  chord Arcs and Chords arc of the chord  -Two minor arcs are iff their corr chords are - Inscribed polygons has each vertex on the circle - If the diameter of a circle is perpendicular to a chord, it bisects the cord & the arc -Two chords are iff they are equidistant from the center. 11 11 .

  5. An inscribed has its vertexon the circle Inscribed polygons have all vertices on the circle Opposite ‘s of inscribed quadrilaterals are supplementary The measure of inscribed ’s = ½ intercepted arc If an inscribed intercepts a semicircle, the = 90° If 2 inscribed ‘s intercept the same arc, the ‘s are Inscribed Angles  Inscribed Intercepted arc red & blue ‘s are

  6. Tangent lines intersect the circle at 1 point—the ‘point of tangency’ A line is tangent to the circle iff it is perpendicular the the radius drawn at that particular point • Tangents • if a point is outside the circle & 2 tangent segments are drawn from it, the 2 segments are congruent. . Tangents can be internal or external •  

  7. A secant line intersects the circle in 2 points I Secants, Tangents & Angle Measures intersecting at point of tangency A B C D Central angles 1 secant & 1 tangent

  8. II Secants, Tangents & Angle Measures intersection in interior of circle B C 2 secants: forms 2 pair of vertical angles – vertical 1 2 A D

  9. III Secants, Tangents & Angle Measures Intersection at exterior point Case 1 2 secants C B P A D C D Case 2 1 secant & 1 tangent P A B Case 3 2 tangents B P Q A

  10. If two chords intersect inside (or outside) of a circle, the products of their segments are equalab = cd 2 secants & exterior point:: a(a + x) = b(b + c) c Special Segments in a Circle b a d x a b c 1 tan and 1 sec & exterior point a a2 = x(x + b) = x2 + bx b x

  11. Point P (h, k) is the center of a circle. Radius of the circle = r Equations of circles y • (h, k) x The equation of this circle: (x – h)2 + (y – k )2 = r 2

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