Circles

Parts of a Circle

Circle A circle is the set of all points in a plane that are a given distance from a given point in the plane, called the center of the circle.

Segments of a Cirlce Radius – has one endpoint on the center and one on the circle Chord – has both endpoint on the circle Diameter – a chord that passes through the center

Theorem 11-1 All radii of a circle are congruent.

Theorem 11-2 The measure of the diameter of a circle is twice the measure of the radius of the circle.

Arcs and Central Angles

Types of Arcs Minor Arc – measure is less than 180° Major Arc – measure is greater than 180° Semicircle – measure equals 180°

Definition of Arc Measure The degree measure of a minor arc is the degree measure of its central angle. The degree measure of a major arc is 360 minus the degree measure of its central angle. The degree measure of a semicircle is 180.

Postulate 11-1 The sum of the measures of two adjacent arcs is the measure of the arc formed by the adjacent arcs.

Theorem 11-3 In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent.

Arcs and Chords

Theorem 11-4 In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

Theorem 11-5 In a circle, a diameter bisects a chord and its arc if and only if it is perpendicular to the chord.

Inscribed Polygons

Inscribed Polygon A polygon is inscribed in a circle if and only if every vertex of the polygon lies on the circle. The circle is circumscribed about the polygon

Theorem 11-6 In a circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center.

Circumference of a Circle

Circumference The distance around a circle

Theorem 11-7 If a circle has a circumference of C units and a radius of r units, then C= 2  r or C =  d.

Area of a Circle

Theorem 11-8 If a circle has an area of A square units and a radius of r units, then A =  r 2

Theorem 11-9 If a sector of a circle has an area of A square units, a central angle measurement of N degrees, and a radius of r units, then A = N/360(  r 2 )