shlo326

=Chapter 9=

Objectives:
Define a circle and its associated parts and use them in constuctions Define and use the degree measure of arcs. Define and use the measure of arcs. Prove a theorem about chords and their intercepted arcs.

Blue Boxes:
[|Cicle vocab help]
 * Circle:** A circle is the set of all points in a plane that are equidistant from a given point in the plane know as the center of a circle.
 * Radius:** (plural:radii) is a segment from the center of the circle to a point on the circle.
 * Chord:** is a segment whose endpoint line on a circle.
 * Diameter:** is a chord that contains the center of a circle.
 * Semicircle:** is an arc whose endpoints are endpoints of a diameter. A semicircle is informally called a half-circle. a semicircle is named by its endpoints and another point that lies on the arc.
 * Central angle:** of a circle is an angle in the plane of a circle whose vertex is the center of the circle.
 * Interceptered arc:** whose endpoints lie on the sides of the angle and whose other points lie in the inteerior of the angle.
 * Degree Measure of Arcs:** The degree measure of a minor arc is the measure of its central angle.The degree measure of a arc is 360 degrees minus the degree measure of its arc. The degree measure of a semicircle is 180 degrees.
 * 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 the circle, or in congruent circles, the arcs of congruent chords are similar.

Example:
Find the measure of arc XY, arc YZ, and arc XYZ. The measures of arc XY and YZ are found from their central angles. measure of arc XY=100° and measure of arc YZ=120° Arc XY and arc YZ, which have just one endpoint in common, are called adjacent arcs. Add their measres to find the measure of arc XYZ. measure of arc XYZ measure of arc XY + measure od arc YZ 100° + 120° = 220°

**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.

**Blue Boxes:**
at two points. which is known as the point of tangency. the circle drawn to the point of tangency. circle the chord. a circle at its endpoint on the circle, then the line is to the circle. [|Cool link]
 * Secants and Tangents: A secant** to a circle is a line that intersects with the circle
 * A tangent** is a line in the plane of the circle that intersects the circle at exactly one point,[[image:Tangents_and_Secants.JPG align="right"]]
 * Tangent theorem:** If a line is tangent to a circle, then the line is to a radius of
 * Radius and Chord Theorem:** A radius that is perpendicular to a chord of a
 * Converse of the Tangent Theorem:** If a line is perpendicular to a radius of
 * Theorem:** The perpendicular bisector of a chord passes through the center of the circle.

Example:
(AX)² + 3² = 5² (AX)² = 5² - 3² (AX)² = 16 AX = 4 Line PR bisects line AB so BX = AX which is 4 So, AB = AX + BX or 4 + 4 equals 8

**Objetives:**
Define inscribed angle and intercepted arc. Develop and use the inscribed angle theorem and its corollaries.

Blue Boxes:
[|sweet link]
 * Inscribed angle theorem:** The measure of an angle inscribed in a circle is equal to 1/2 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.
 * Inscribed angle:** an angle whose vertex lies on a circle and whose sides are chords of the circle.[[image:9.2_circle.JPG align="right"]]

Example:
Angle ABC is inscribed in circle P and intercepts arc AC. By the Inscribed Angle Theorem: measure of angle ABC is ½ the measure of arc AC = ½ (90°) equals 45°

**Objectives:**
Define angles formed by secants and tangents of circles. 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. [|Awsome link]

**Blue Boxes:**
that intersect in the exterior of a circle is 1/2 the different of the measure of the intercepted arcs.
 * 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 ½ the measure of its intercepted arc.
 * 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

Example:
Find angle XYZ in each figure. __Circle A:__ Angle XYZ is formed by a secant and a tangent that intersect on the circle. measure of angle XYZ is ½ measure of arc XY (200°) = 100° __Circle B:__ Angle XYZ is formed by two secants that intersect inside the circle. measure of angle XYZ is ½ (measure of arc XY + measure of arc PQ)1/2 (100° + 20°) = 60° __Circle C:__ Angle XYZ is formed by two secants that intersect outside the circle. measure of angle XYZ is ½ (measure of arc XZ - measure of arc PQ)1/2 (100° - 50°) =25°

**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.

**Blue Boxes:**

 * 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.
 * 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 on the product of the lenghts of the segment of one chord equals the product of the length of the chord.

Example:
__Circle A:__ XY = XZ because they are tangents to a circle from the same external point. __Circle B:__ Whole x outside = Whole x outside so AC•BC = EC•DC 8•3 = 6•DC DC = 4 __Circle C:__ Let //d// be the distance to the end. //d// • 50 = 100² //d •// 50 = 10,000

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

Example:[[image:9.6_circle.JPG align="right"]]
Given: x²+y²=81 To find the center and radius of this circle we have to find the square root of 81. Which is 9 and that is our radius. Since the center is at (0,0) (i found that out because x and y dont have any numbers before them). Next you graph it. Your center is at (0,0) and your radius is 9. So on the x and y axsis extend out.