Wisa27

=Chapter 9 Circles=

Objectives:

 * Define a circle and its associated perts,and use them in construction.
 * Define and use the degree measure of arcs.
 * Define and use the length measure of arcs.
 * Prove the theorem about chords and their intercepted arcs.

Definition: 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 a 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.

Definitions: Central Angle and Intercepting Arc
A **central angle** of a circle is an angle in the plane of a circle whose vertex is the center of 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 **intercepted arc** of the central angle.

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

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(2pie//r//)

Chords and Arcs Theorem
In a circle, or in congruent circles, the arcs of congruent chords are congruent

The Converse of the chords and Arcs Theorem
In the circle or in congruent circles, the chords of congruent arcs are congruent =Chapter 9 Circles=

Objectives:

 * Define tangents and secants of circles.
 * Understand the relationship between tangents and certain radii of circles.
 * Understand the geometry of a radius perpendicular to a chord of a circle.

Secants and Tangents
A **secant** to a circle is a line that intersects the circle at two points. A **tangents** is a line in the plane of the 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 Theorem
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.

=Chapter 9 Circles= ====

Objectives:

 * Define and inscribed angle and intercepted arc.
 * Develop and use the inscribed Angle Theorem and its corollaries.

Inscribed Angle Theorem
The measure of an angle inscribed in a circle is equal to the measure of the intercepted arc.

Right-Angle Corollary
If an inscribed angle intercepts a semicircle, then the angle is a right angle.

Arc-Intercepting Corollary
If two inscribed angles intercept the same arc, then they have the same measure.

=Chapter 9 Circle=

Objectives:

 * Define angles formed by secants and tangents of circles.
 * Develop and use theorems about measure of arcs intercepted by these angles.

Theorem
If a tangent and a secant (or a chord) intersects on a circle at the point of tangency, then the measure of the angle formed is the measure of its intercepted arc.

Theorem
The measure of an angle formed by two secants or chords that intercepts in the interior of a circle is the of the measure 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 the of the measure of the intercepted arcs.

Theorem
The measure if a secant-tangent with it vertex outside the circle is one-half the difference to the measures if 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 degrees.

=Chapter 9 Circle=

Objectives:

 * Define special cases of segments related to circles, including secant-secant, secant-tangent, and chord-chord segments.
 * Develop and use theorems about measures of the segments.

Theorem
If two segments are tangent to a circle from the same external point, then the segments are of equal length.

Theorem
If two secants intersect outside the circle, the product of the lenghts of one segment and its external segment equals the product of the lengths to the other secant segment and its external segment.

Theorem
If the secant and a tangent intersects outside the circle, then the product of the lengths of the secants segment and its external segment equals the length of the tangent segment squared.

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.

=Chapter 9 Circles=

objectives:

 * Develop and use the equation for a circle.
 * Adjust the equation for a circle to move the center in a coordinate plane.