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=Chapter 9=

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

Vocabulary
Circle- set of all points in a plane that are equidistant from a given point in the plane known as the center of the circle Radius- segment from the center of the circle to a point on the circle Chord- segment whose endpoints line on a circle Diameter- chord that contains the center of the circle Arc- unbroken part of a circle Endpoints- two distinct points on a circle that divide the circle into two arcs Semicircle- the arc of a circle whose endpoints are the endpoints of a diameter Minor Arc- arc of a circle that is shorter than a semicircle of that circle Major Arc- arc of a circle that is longer than a semicircle of that circle Central Angle- angle on the plane of a circle whose vertex is the center of the circle Intercepted Arc- arc whose endpoints lie on the sides of an inscribed angle

Formulas
Arc length- L=M÷360°x(2×3.14r)

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

Example
Find the arc length of the bicycle wheel when the radius is 180 mm and the measure of an arc is 22.5°

[[image:wheelbic.jpg width="192" height="175" align="left" link="http://www.flickr.com/photos/16869306@N00/494054934/"]]
You can change the measure arc to 1/16 since there are 16 sectors on the wheel L=1/16(2x3.14r) L=1/16(2x565.2) L=1/16(1130.4) L=70.65 mm

Objectives
-Define tangents and secants of circles -Understand the relationship between tangents and certain radii of circles -Understand the geometry of a radius prependicular to a chord of a circle

Vocabulary
Secant- line that intersects the circle at two points Tangent- line in the plane of the circle that intersects the circle at exactky one point Point of tangency- the point of intersection of a circle or sphere with a tangent line or plane TheoremsTangent 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 a Tangent Theorem- If a line perpendicular to a radius of a circle at its endpoints on the circle, then the line is tangent to the circle Theorem 9.2.5- The perpendicular bisector of a chord passes through the center of the circle

Example
Find the length of BD. (BE)²+3²=5² (BE)²=5²-3² (BE)²=16 BE=4

Objectives
-Define inscribed angle and intercepted arc -Develop and use the inscribed Angle Theorem and its corollaries

Vocabulary
Inscribed Angle- angle whose vertex lies on a circle and whose sides are chords of the circle

Theorems
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 and 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

Example
Find the measure of the angle when the arc measure is 45° m<=xvy=1/2mXY=1/2(45)=22 1/2

Objectives
-Define angles formed by secants and tangents of circles -Develop and use theorems about measures of arcs intercepted by these angles

Classification of Angles with Circles
Case 1:Vertex is on the circle

Theorems
Theorem 9.4.1-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 one-half the measure of its intercepted arc Theorem 9.4.2-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 measure of the arcs intercepted by the angle and its vertical angle Theorem 9.4.3- The measure of an angle formed by two secants that intersect in the exterior of a circle is one-half the difference of the measure of the intercepted arcs Theorem 9.4.4-The measure of a secant-tangent angle with its vertex outside the circle is one-half the difference of the measure of the intercepted arcs Theorem 9.4.5-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°

Example
A circle is intersected by a tangent and a secant, which forms the angle BFD. The minor chord that it intersects is 180 degrees. Find the measure of angle BFD. Solving: To find the measure of BFD you divide the chord it intersects by 2. Answer: 180/2=90. ABC is 90 degrees.

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

Vocabulary
Tangent segment-a segment that is contained by a line tangent to a circle and has one of the endpoints on the circle Secant segment-a segment that contains a chord of a circle and has one endpoint interior to the circle and the other endpoint 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

Theorems
Theorem 9.5.1-If two segments are tangent to a circle from the same external point, then the segments are of equal length Theorem 9.5.2-If two secants intersect outside a 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 9.5.3-If a secant and a tangent intersect outside a circle, then the product of the lenghts of the secant segment and its external segment equals the length of the tangent segment squared. (Whole x Outside= Tangent Squared) Theorem 9.5.4-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.

Example
The span of a metal detector is 10 feet. IO, is 10 feet, what is JO? Solving: IO and JO are tangents to a circle from the same external point. This means that they are equal.

Answer: IO feet

Objectives
-Develop and use the equaiton of a circle -Adjust the equation for a circle to move the center in a coordinate plane

Formulas
Moving the center of the circle- (x-h)²+(y-k)²=r² Deriving the Equation of a Circle- x²+y²=r²

Example
Find the center of the circle and the radius (x-7)²+(y+3)²=36 use the standard form of the equation (x-h)²+(y-k)²=r² to be replaced with the given equation (x-7)²+(y+3)²=36 therefore, h=7, K=3, and r=6 using opposite reciprocals, -7 is 7 and 3 is -3 center is (7,-3) and radius is 6 units