toga325

chapter 9

=__9.1 Chords and Arcs__=

Objectives

 * define circle and a its associated parts, and use them in constructions.
 * define circle and a its associated parts, and use them in constructions.


 * define and use the degree measure of arcs.


 * define and use the length measuer of arcs.


 * prove a theorem about chords thier intercepted arcs.

= =

__Blue boxes__

 * Major arc:** a circle is an arc that is longer than a semicircle of that circle.
 * Raduis:** a segment from the center of the circle to a point on the circle.
 * Chord:** a segment whose endpoints line on a circle.
 * Arc:** an unbroken part of a circle.
 * Semicircle:** an arc whose endpoints are endpoins of a diameter.
 * Minor arc:** a circle is an arc that is longer than a semicircle of that circle.
 * Diameter:** a chord that contains the center of a circle.
 * Endpoints:** a point at an end of a segment or the starting point of a ray.

__Definitons: central angle and intercepted arc.__
A **central angle** of a circle is an angle in the plane of a circle of a circle whose vertex is the center of the circle. An arc whose endpoitns 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 arcs:__
the degree measure of a minor arc is the measure of its central angle. the degree measusre 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(2phir)

__Chords and Arcs Theorem:__
In a circle, or in congruent circles, the arcs of congruent chords are equal.

= =

__The converse of the chords and arcs theorem:__
In a circle or in congruent circles, the chords of congruent arcs are equal.

=**__Example:__**= On a dartboard the radius is 170mm. If there are 20 sections on the board find the arc of one section. lenght of the arc 1/20. C=2(phi)r length=1/20(2phi170) lenght of the arc is 53mm.

=**__9.2__**= Objectives: 1. Define tangents and secangts of circles.
 * 2.** Understand the relationship between tangents and certain radii of circles.
 * 3.** 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 **tangent** is a line in the plane of the circle that intersects the circle at exactly one poin, which is known as the **point of tangency.**
 * Blue Boxes:**

__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 endpoints on hte circle then the line is tangent to the circle.

__Theorem__ The perpendiculat bisector of a chord passes though the center of the circle.

Circle P has a radius of 6in. and PX is 2in. Line PR is perpendicular to Line AB at point X. Find AB. pythagorean theorem: (AX)^2+2^2=6^2 (AX)^2=6^2-2^2 (AX)^2=32 AX=16 9.3__ Objetives: 1.** Define inscribed angle and intercepted arc.
 * Example:**
 * __[[image:djkhdhgyyeyyey.JPG]]
 * 2.** Develop and use the inscribed angle theorem and its corollaries.


 * Inscribed angle:** an angle whose vertex lies on a circle and whose sides are chords of the circle.

__Inscribed angle theorem__ the measure of an angle inscribed in a circle is equal to 1/2 the measure of the intercepted arc.
 * Blue Boxes:**

__Right Angle Corollary__ If an inscribed angle intercepts 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.

Find the measure of <XVY. <XVY is inscribed in P and intercepts arc XY. m<XVy (1/2)m(arc)XY (1/2)(45) = 221/2 Objectives: 1.** Define angles formed by secants and tangents of circles.
 * Example:**
 * __9.4__
 * 2.** Develop and use theorems about measures of arcs intercepted by these angles.


 * 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 circle at the point of tangency, then the measure of the angle formed is 1/2 the measure of its intercepted arc.
 * Blue Boxes:**

__Theorem:__ The measure of an angle formed by two secants or chords that intersect in the interior of a circle is 1/2 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 1/2 the different of the measure of the intercepted arcs.

Find m<AVC in each figure. A. B. C.
 * Examples:**

a. <AVC is formed by a secant and a tangent that intersect on the circle. m<AVC (1/2)m(arc)AV (1/2)(150) = 75. b. <AVE is formed by two secants that intersect inside the circle. m<AVC (1/2)(m arcAC + m arcBD) (1/2)(80 +40) = 60 c. <AVC is formed by two secants that intersect outside the circle. m<AVC (1/2)(m arcAC - m arcBD) (1/2)(80 - 20) = 30.

Objectives: 1.** Define special cases of segments related to circles, including secant-secant, secant-tangent and chord-chord segments.
 * __9.5__
 * 2.** Develop and use theorems about measures of the segments.


 * Tangent Segment:** a segment that is contained by a line tangent to a circle and has one of its endpoints on the circle.
 * Secant Segment:** a segment that contains a chord of a circle and has one endpoint exterior to the circle and the other endpoints on the circle.
 * External Secant Segment:** the portion of a secant segment that lies outside the circle.
 * Chord:** a segment whose endpoints lie on a circle.

__Theorem:__ If two segments are tangent to a circle from the same external points, then the segments are equal.
 * Blue Boxes:**

__Theorem:__ If two secants intersect outside a circle, the product of the lengths of one secant segment and its external segment equals the prroduct of the lengths of the one secant, (whole * outside = whole * outside)

__Theorem:__ If a secant and a tangent intersect outside a circle, then the product of the lenghts of the segment and its external segment equals tangent squared. ( whole * outside = tangent squared)

__Theorem:__ If two chords intersect inside a circle the nthe product of the lenghts of the segment of one chord equals the product of the length of the chord..


 * Examples:**

Objectives: 1.** Develop and use the equation of a circle.
 * __9.6__
 * 2.** Adjust the equation for a circle to move the center in a coordinate plane.