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=CHAPTER 9=

[[image:window487944510_6e7ff0c7e7.jpg width="227" height="215" align="left" link="http://www.flickr.com/photos/two375/487944510/"]]SECTION 1
OBJECTIVES ~Define a circle and its associated parts, anad 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.

In the picture to the left, the blue line is an example of a diameter, the yellow line is an example of a chord, the pink line is an example of a radius. VOCABULARY ~Circle: The set of points in a plane that are equidistant from a given point known as the center of the circle. ~Arc: An unbroken part of a circle *an example is "("* ~Semicircle: The arc of a circle that measures 180°. ~Minor Arc: An arc of a circle that measures 179° or less. Described with 2 letters. ~Major Arc: An arc of a circle that measures 181° or more. Described with 3 letters. ~Central Angle: An angle formed by 2 rays originating from the center of the circle. ~Intercepted Angle: The arc between the the rays of the central angle. ~Arc Length: The measure of the arc of a circle in terms of linear units, such as centimeters. ~Arc Measure: The measure of an arc in a circle in terms of degrees.

THEOREMS AND FORMULAS ~Cords and Arcs Theorem: In a circle, or in congruent circles, the arcs of congruent chords are congruent. ~Converse of the Cords and Arcs Theorem: In a circle, or in congruent circles, the chords of congruent arcs are congruent. ~Arc Length: L=M/360(2piR) *L-length, M-degree measure of arc, R-radius of circle*

SOLVING PROBLEMS Determine the length of an arc with the given central angle (m<D) and radius (r); Round answers to the nearest hundreth. 1: m<D: 65°; r: 6 L=M/360(2piR) L=65/360(2pi6) L=.10856(2pi6) L=6.81 Determine the degree measure of an arc with the given length (L) and radius (r) 2: L: 56; r: 30 L=M/360(2piR) 56=M/360(2pi30) 56=M/360(188.4955592) ÷ each side by 360 .15556=M(188.4955592) × each side by 188.4955592 M=29.32

[[image:ferris92802870_daf251679c.jpg width="274" height="186" align="left" link="http://http//www.flickr.com/photos/maltloafer/92802870/"]]SECTION 2
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.

The picture to the left shows a tangent line to a circle. The yellow arrow shows the point of tangency.

VOCABULARY ~Secant: A line that intersects a circle at 2 points. ~Tangent: A line that intersects a circle at 1 point. ~Point of Tangency: The point of intersection of a circle with a tangent line.

THEOREMS AND FORMULAS ~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 9.2.5: The perpendicular bisector of a chord passes through the center of the circle.

SOLVING PROBLEMS Answer the following questions by using the circle at the left. 1: If AB equals 4 and AE equals 1, what is AD? What is EB? AB and AD are congruent so AD also equals 4 AE, AB and EB form a right triangle so you can figure these out by using the pythagreon theorem (A²+B²=C²). 1²+B²=4² 1+B²=16 -1 from each side B²=15 Take the square root of 15 EB=3.87

[[image:clock16734948_73cbe09dfe.jpg width="288" height="273" align="left" link="http://www.flickr.com/photos/simpologist/16734948/"]]SECTION 3
OBJECTIVES ~Define inscribed angle and intercepted arc. ~Develop and use the Inscribed Angle Theorem and its corollaries.

The picture at the left shows an example of inscribed angles.

VOCABULARY //none//

THEOREMS AND FORMULAS ~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 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.

SOLVING PROBLEMS Find the following information using the circle to the left. m<LPJ: 30° and m<KMJ: 45° 1: m<LPM <LPM=150° because it is in a linear pair with <LPJ

2:measure of arc ML measure of arc ML=150 because it is the intercepted arc of the central angle LPM

[[image:grape347005873_37285ccd93.jpg width="265" height="216" align="left" link="http://www.flickr.com/photos/sonykus/347005873/"]]SECTION 4
OBJECTIVES ~Define angles formed by secants and tangents of circles. ~Develop and use theorems about measures of arcs intercepted by these angles.

The picture at the left is an example of an angle formed by a secant and a tangent. Line CE is tangent to circle A at C while line BD is secant. The angle formed is angle CED.

VOCABULARY //none//

THEOREMS AND FORMULAS ~Theorem 9.4.1: If a tangent and a secant (or chord) intersect on a circle at the point of tangency, then the measure of the angle formed in one-half the measure of its intercepted arc. (intercepted arc÷2=angle measure) ~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 measures 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 measures 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 measures 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°

SOLVING PROBLEMS Find the following information if the measure of arc EF equals 20°, the measure of arc FH equals 180°, the measure of arc DH equals 45°, and line GD is tangent to circle T at D. 1: m<FDG, m<GFD and m<FGD m<FDG=90° because of theorem 9.4.1 m<GFD=12.5° because of theorem 9.4. m<FGD=77.5° because all triangles are 180° so 180-90-12.5 is 77.5

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[[image:soccer427717795_47d2aeeec1.jpg width="283" height="211" align="left" link="http://www.flickr.com/photos/benloveridge/427717795/"]]SECTION 5
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.

A soccer ball is another example of a circular object.

VOCABULARY //none//

THEOREMS AND FORMULAS ~Theorem 9.5.1: If two segments are tangent to a circle form the same external point, then the segments are of equal length. ~Theorem 9.5.2: If two secants intersect the outside of a circle, then the product of the lengths of the 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 lengths 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.

SOLVING PROBLEMS 1: Find X in the circle to the left. 5X=6*10 5X=60 ÷ each side by 5 X=12 because of theorem 9.5.4

~*~ [[image:sand_dollar357006813_42a9b9bc28.jpg width="333" height="215" align="left" link="http://www.flickr.com/photos/my3sons_nh/357006813/"]]SECTION 6
OBJECTIVES ~Develop and use the equation of a circle. ~Adjust the equation for a circle to move the center in a coordinate plane.

A sand dollar is another circle.

VOCABULARY //none//

THEOREMS AND FORMULAS ~X²+Y²=R²: To find the X and Y intercepts of a circle whose center is the origin (0,0). ~(X-H)²+(Y-K)²=R²: To find the X and Y intercept of a circle whose center is NOT the origin.

SOLVING PROBLEMS 1: Find the X and Y intercepts of the circle to the left. The radius is 1.75 so the intercepts will be (0,-1.75), (0,1.75), (-1.75,0) and (1.75,0)

~*~

EXAMPLES
Section 1: Determine the length of an arc with the given central angle measure (m<H) and radius (r); Round answers to the nearest hundreth.

1: m<H: 40°; r: 6 2: m<H: 118°; r: 30

Section 2: Refer to the circle to the left to answer the questions.

3: If AB equals 5 and EA equals 3, what is AD? What is CE?

4: If AE is 9 and EB is 12, what is AB?

Section 3: In circle P, m<LPJ: 30° and m<KMJ: 45°. Find the following information.

5: What is m<LPM?

6: What is the measure of arc LJ?

Section 4: Use the circle at the left to answer the following questions. The measure of arc BD is 80°, m<FAD=35°, and EC is tangent to circle T at D and they form a right angle. 7: m<FAB

8: m<BCD

Section 5:

9: Find X in the circle to the left.

Section 6:

10: Find the center and radius for the circle to the left.

11:Right an equation for the circle to the left.

ADDITIONAL HELP
[|Click Here] for video help on arcs and chords. Under the Video (Flash) Lessons column, click on Arcs and Chords to view the video. [|Click Here] for help with tangents, secants, and more. [|Click Here] for help in the covered areas but also circumference and area of circles. [|Click Here] to see different job oppurtunities that involves geometry.

ANSWERS
1: 4.2 2: 61.8 3: AD=5 because it is congruent to AB and they are both the radius of the circle. CE=2 because AC is also a radius so you take AC (5) - AE (3). 4: AB: 15 because AE, AB, and EB form a right triangle so you can use the Pythagreon Theorem (A²+B²C²) to figure it out. 5: <LPM and <LPJ form a linear pair so take 180-<LPJ and you get **150°** 6: Arc LJ=30 because <LPJ is a central angle and the intercepted arc is LJ. 7: 75° 8: 50° 9: 2.7 10: center: (1,2) radius: 2 11: (X-1)²+(Y-2)²=4


 * ALL EXAMPLE CIRCLES WERE DRAWN BY ME****