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

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

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

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.

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 

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  .

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

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

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

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

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