hare215

=Circles=

Chords and Arcs
A **radius** (plural, radii) is a segment from the center of the circle to a point on the circle. A **chord** is a segment whose endpoints line on a circle. A **diameter** is a chord that contains the center of a circle. the circle. An arc whose endpoints lie on the sides of the angle and whose other points lie in the interior of the angle is the arc is 360° minus the degree measure of its minor arc. The degree measure of a semi-circle is 180°. the following - **L = M / 360° (2pi//r//)** Semi-circle - arc of a circle whose endpoins are the endpoints of the diameter Major Arc - an arc that is longer than a semi-circle, more than 180°. Minor Arc - an arc that is not longer than a semi-circle, less than 180°.
 * __Circle__:** A **circle** is the set of all points in a plane that are equiditant from a given point in the plane known as the center of the circle.
 * __Central Angle and Intercepted Arc__:** A **central angle** of a circle is an angle in the plane of a circle whose vertex is the center of
 * intercepted arc** of the central angle.
 * __Degree Measure of Arcs__:** The degree measure of a minor arc is the measure of its central angle.The degree measure of a major
 * __Arc Length__:** If //r// is the radius of a circle and M is the degree measure of an arc of the circle, then the length, L, of the arc is given by
 * __Chords and Arc Theorem__:** In a circle, or in congruent circles, the arcs of congruent chords are congruent
 * Vocabulary**

Tangents to Cicrles
circle that intersects the circle at exactly one point, which is known as the **point of tangency**. tangency. tangent to the circle.
 * __Secants and Tangents__:** A **secant** to a circle is a line that intersects the circle at two points. A **tangent** is a line in a plane of a
 * __Tangent Theorem__:** If a line is tangent to a circle, then the line is perpendicular to a radius of the circle drawn to the point of
 * __Radius and Chord Theorem__:** A radius that is perpendicular to a chord of a circle bisects the chord.
 * __Converse of the Tangent Theorem__:** If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is
 * __Theorem__:** The perpendicular bisector of a chord passes through the center of the circle.

Inscribed Angles and Arcs

 * __Inscribed Angle Theorem__:** The measure of an angle inscribed in a circle is equal to one-half the measure of the intercepted arc.
 * __Right Angle Corollary__:** If an inscribed angle intersects a semicircle, then the angle is a right angle.
 * __Arc-Intercept Corollary__:** If two inscribed angles intercept the same arc, then they have the same measure.

Angles Formed by Secants and Tangents

 * __Classification of Angles with Circles__:**
 * Case 1** - Vertex is on the circle
 * Case 2** - Vertex is inside the circle
 * Case 3** - Vertex is outside the circle
 * __Theorem__:** If a tangent and a secant (or a chord) intersect on a circleat the point of tangency, then the measure of the angle formed is one-half the measure of its intercepted arc.
 * __Theorem__:** The measure of an angle formed by two secants or chords that intersect in the interior of a circle is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
 * __Theorem__:** The measure of an angle formed by two secants that intersect in the exterior of a circle is one-half the difference of the measures of the intercepted arcs.
 * __Theorem__:** The measure of a secant-tangent angle with its vertex outside the circle is one-half the difference of the measures of the intercepted arcs.
 * __Theorem__:** The measure of a tangent-tangent angle with its vertex outside the circle is one-half the difference of the measures of the intercepted arcs, or the measure of the major arc minus 180°.

Segments of Tangents, Secants and Chords

 * __Theorem__:** If two segments are tangent to a circle from the same external point, then the segments are of equal length.
 * __Theorem__:** If two segments intersect outside the circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment. (Whole x Outside = Whole x Outside
 * __Theorem__:** If a secant and a tangent intersect outside a circle, then the product of the lengths of the secant segment and its external segment equals the length of the tangent segment squared. (Whole x Outside = Tangent²)
 * __Theorem__:** If two chords intersect inside a circle, then the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.

Circles in the Coordinate Planes
The equation of a circle with the center at (0,0) is: x² + y² = r², where x and y equal (x,y) and r equals radius. Example Picture The equation of a circle with the center not on (0,0) is: (x-h)² + (y-k)² = r², where h and k equal the center, (h,k). Example Picture Example Questions (with answers) [|Circle Game] [|Page with tons of Geometery information; scroll down some for circles]