sira15

http://www.flickr.com/photos/33732557@N00/363989684/ • Define a circle and its associated parts, and sue 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.
 * CHAPTER 9 SECTION 1 CHORDS AND ARCS**
 * Objectives**:

Circle - the set if all points in a plane that are equidstant from a given point in the plane known as the center. Radius - is a segment from the center of the circle to a point on the circle. (plural radii). Chord - is a segment whose endpoints line on a circle. Diameter - is a chord that contains the center of a circle. Central Angle - of a circle is an angle in the plane of a circle whose vertex is the center of the circle. Intercepted Arc - an arc whose endpoints lie on the sides of the angle and whose other points lie in the interior of the angle. Degree Measure of Arcs: - of minor arc is the measure of its central angle. - of a major arc is 360° minus the degree measure of its minor arc. - The degree measure of a semi circle is 180°. Arc Length- L= M/360° (2pi•r)
 * Definitions**:

Chords and Arcs Theorm- In a cirlce, or in comgruent cirles, 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 congruents arcs are congruent.
 * Theorems** -

• **Example**: Find the measures of arcs BD, DC, and BDC //Solution// - The measures of arc BD and arc DC are fond from their central angles. The measure of arc BD= 50º and measure of arc DC= 40º Arc BD and arc DC, which have just one endpoint in common, are called adjacent arcs. Add their measure to find the measure of arc BDC. m of arc BDC m of arc BD + m of arc DC 50º + 40º = 90º



http://www.flickr.com/photos/88261881@N00/365455942/
 * CHAPTER 9 SECTION 2 TANGENTS TO CIRCLES**

• Define tangent 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
 * Ojectives**:

Secants and Tangents: A secant to a circle - a line thta intersects the circle at two points. A tangent - a line in the plane of the circle that intersects the circle at exactly one point (point of tangency)
 * Definitions**:

If a line is tangent to a circle, then the line is perpendicular to a radius to a radius of the circle drawn to the point of tangency. A radius that is perpendicular to a chord of a circle intersects the chord. If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is parallel. The perpendicular bisector of a chord passes through the center of the circle.
 * Theorems**:
 * •**Tangent Theorm-
 * •**Radius and Chord Theorem-
 * •** Converse of the Tangent Theorem**-**
 * •** Theorem-

Find AB. //Solution// By the Pythagorean Theorem - (AX)² + 8² = 10² (AX)² = 10² - 8² (AX)² = 36 AX = 6 By the Radius and Chord Theorem, PR bisects AB, so BZ = AZ = 6. Therefore, AB = AZ + BZ =6+6 =12
 * Example**: C has a radius of 8n. and CZ is 6in. Radius CD is perpendicular to chord AB at point Z

http://www.flickr.com/search/?q=Inscribed+Angles+&m=text
 * CHAPTER 9 SECTION 3 INSCRIBED ANGLES AND ARCS**

• Define inscribed angle and intercepted arc. • Develop and use the Inscribed Angle Theorem and its corollaries. • Inscribed Angle THM - The measure of an angle inscribed in a circle is equal to ? the measure of the intercepted arc.
 * Objectives**:
 * Theorems**:

• 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.
 * Corollaries:**

Find the measure of angle XVY. //Solution// **-** Angle XVY is inscribed in circle P and intercepts arc XY. By the Inscribed Angle Thm. m angle XVY = ½•m arc XY ½ (50º)25º
 * Example:**

http://www.mathwarehouse.com/geometry/circle/images/two-secants/picture-intersection-of-secants.gif • Define anlges formed by secants and tangents of circles. • Develope and use theorems about measures of arcs intercepted by these angles.
 * CHAPTER 9 SECTION 4 ANGLES FORMED BY SECANTS AND TANGENTS**
 * 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 ? the measure of its intercepted arc. • The measure of an angle formed by 2 secants or chords that intersect in the interior of a circle is ? the ? of the measures of arcs inercepted by the angle and its vertical angle. • The measure of an angle formed by an angle formed by 2 secants that intersect in the exterior of a circle is ? the ? of the measures of the intercepted arcs •the measure of a tangent-tangent angle with its vertex outside the circle is ?????????? Find the measure of angle AVC in each Figure. //Solution// - a.) Angle AVC is formed b a secant and a tangent that intersect on the circle. m ang.AVC = ½ m arc AV ½ (72º) = 36º b.) Angle AVC is formed by two secants that intersect inside the circle. M of ang. AVC = ½ (m arc ACm arc BD) = ½ (100º + 20º)60º c.) Angle AVC is formed by two secants taht intersect outside the circle. m angle AVC = ½ (m arc AC - m arc BD)½ (100º - 50º) = 25º
 * Theorms**:
 * Example**:



http://img.timeinc.net/bmx/content/images/murray_jam_04_06/loop05.jpg
 * CHAPTER 9 SECTION 5 SEGMENTS OF TANGENTS, SECANTS, AND CHORDS**
 * 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.

• If two segments are tangent ot a circle from the same external point, then the segments ?? • If two secants intersect outside a circle, the product of the lengths of one secant segment and its external segment equals ??. (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 equals ??. (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 ??.
 * Theorem**:

5, GX = 2, and FX = 6. Find HX. //Solution// - Line EX and line FX are secants that intersect outside the circle. Use Whole X Outside = Whole X Outside. EX • GX = FX • HX 5 • 2 = 6 • HX 6 • HX = 10 HX =
 * Example**:



http://www.lassp.cornell.edu/LASSPTools/pictures/XYinput.gif • Develop and use the equation of a circle. • Adjust the equation for a circle to move the center in a coordinate plane.
 * CHAPTER 9 SECTION 6 CIRCLES IN THE COORDINATE PLANE**
 * Objectives**:


 * Example**