spsh516

= = =__Chapter 9__= =**9.1**=

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

//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(2pir))

//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 a circle or in congruent circles, the chords of congruent arcs are congruent.

__Example__
The window has a radius of 10 inches, the degree of one of its arcs is 10 degrees. What is the length of the window?

L=1.75
 * L=10/360(2π10)

Vocabulary = = = =

=**9.2**=

Objectives
 * Define tangents and secants of circles.
 * Understand the relationship between tangents and certain radii of circles.
 * Understand the geometry of a riadius 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 tangent 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.

//Tangeant Theorem//
If a line is tangent to a circle, then the line is tangent to a radius of the circle drawn to the point.

//Radius and Chord Theorem//
A radius that is perpendicular to a chord of a circle is equal to the chord.

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

__Example__
A bubble has two perpendicular lines intersecting at its center. Half of one of the lines measures 20cm and the intersecting half of the other line measures 15cm. If you a line was drawn from the top point of one line to the side point of the intersecting line then it would form a right triangle. Find the measure of the chord. Use the pythagorean theorem: 20²+15²=?² 400+225=?² The measure of the chord is 25.** Vocabulary
 * ?=25

=9.3=

Objectives
 * Define 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 length the measure of the intercepted arc.

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

//Arc-Intercept Corollary//
If two insrcibed angles intercep the same arc, then they have the same measure.

Vocabulary =9.4= Objectives
 * Define angles formed bu secants and tangents of circles.
 * Develop and use theorems about measures of arcs intercepted by these angles.

//Theorems//
Vocabulary =9.5= Objectives
 * If a tangent and a secant (or a chord) intersect on a circle at the point of tangency, then the measure of the angle formed is half the measure of its intercepted arc.
 * The measure of an angle is formed by tow secants or chords that intersect in the interior of a circle is half the sum of the measures of the arcs intercepted by the angle and its verticle angle. x1+x2/2
 * The measure of an angle formed by two secants that intersect in the exterior of a circle is half the difference the measures of the intercepted arcs.
 * Define specieal cases of segments related to circles, including secant-secant, secant-tangent, and chord-chord segments.
 * Develop and use theorems about measures of the segmants.

//Theorems//
Vocabulary =9.6= Objectives Vocabulary
 * If two segments are tangent to a circle form the same external point, then the segments equal.
 * If two secants intersect outside a circle, then the product of the lengths of one secant segment and its external segmant equals the product of the lengths of the other secant segment and its external segment. (Whole x Outside = Whole x Outside)
 * 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 = Tangent Squared)
 * 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.
 * Develop and use the equation of a circle.
 * Adjust the equation for a circle to move the center in a coordinate plane.