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=Chords and Arcs=

__Circle__
A circle is the set of all points in a plane that are equidistant from a given point in the plane known as the center of the circle. A **radius** is a segment from the center of the circle to a point on the circle a is a chord that contains the center of a circle.
 * chord** is a segment whose endpoints lines on a circle a **diameter**

circle link. [|Learning site]

__Central Angles and Intercepted Arc__
A **central angle** of plane a circle is an angle in the plane of a circle whose vertex is the center of a 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
 * intercepted arc** of the central angle.

__Degree Measure of Arcs__
The degree measure of a minor arc is the mesure of its central angle. The degree measure of a major arc is 360° minus the degree measure of its minor arc. The degree measure of a semicircle is 180°.

__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 the following:
 * L=M/360°(2//π//r)**

__Chords and Arcs Theorems__
In a circle, or in congruent circles, the arcs of congruent chords are congruent.

__The converse of the chords and arcs theorem__
In a circle or in congruent circles, the chords of congruent arcs are congruent.

=Tangents to Circles=



**__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 circle that intersects the circle at exactly one point, which is known as the **point of tangency**.

__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 tangency.

__Radius and Chord Theorem__
A radius that is perpendicular to a chord of a circle bisects the chord.

Converse of the Tangent
If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.

__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 Plane=



__Circle Equation__
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.//

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).//

**//Examples!//**

[|good learning site] [|learning tool for tangents and secants.]