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Circles Vocabulary Unit 7 OBJECTIVES: Degree & linear measure of arcs Measures of angles in circles Properties of chords, tangents, & secants.

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Presentation on theme: "Circles Vocabulary Unit 7 OBJECTIVES: Degree & linear measure of arcs Measures of angles in circles Properties of chords, tangents, & secants."— Presentation transcript:

1 Circles Vocabulary Unit 7 OBJECTIVES: Degree & linear measure of arcs Measures of angles in circles Properties of chords, tangents, & secants

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 = π d = 2π r Concentric circles share the same center & have different radius lengths

3 Angles and Arcs Measure 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 Length is the proportion of the circumference formed by the central angle : L T V.V. P

4 Arcs and Chords -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. arc of the chord   chord 11.

5 Inscribed Angles An inscribed has its vertex on 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  red & blue ‘s are  Inscribed Intercepted arc 

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

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

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

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

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


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