chapter+9+!!!!!!!!!!

Definitions Circle- A circle is the set of all points in a plane that are equidstant from a given point in the plane know as the center of the circle.
 * 9.1 Chords and Arcs**

Radius- The radius is a segment from the center of the circle to a point on the circle

Chord- A chord is a segment whose endpoints lay on s circle

Diameter- A diameter is a chord that contains the center of a circle

Arc- An arc is a unbroken part of a circle

Endpoints- Any two distinct points on a circle divide the circle into two arcs

Semi circle- A semi circle is an arc whose endpoints are the endpoints of a diameter

Minor arc- A minor arc of a circle is an arce that is shorter than a semicircle of that circle

Major arc- A major arc of a circle is an arc that is longer than a semi circle of that circle

Central angle- A 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 in the interior of the angle

Degree measure of arcs- The gegree measure of a minor are is the measure of its central angle. The degree measure of a major arc is 360 degrees minus the degree measure of its minor arc. The degree measure of a semicircle is 180 degrees.

DEGREE MEASURE OF ARCS- *The degree measure of a minor arc is tha measure of its central angle.
 * The degree measure of a major arcis 360 degrees minus the degrre measure of its minor 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 degrees(2 pie r) CHORDS AND ARCS THEOREM- In a circle, or in congruent circles, 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 congruent arcs are __congruent.__

now for some real life examples!!!!



this a stadard NBA basketball court as you can see, a lot of it is based on arcs!

__**DEFINITIONS**__
 * 9.2**

SECANT- Is a line that intersects the circle at two points. TANGENT- Is a line in the plane of the circle that intersects the circle at exactly one point. POINT OF TANGENCY- Is the one point that the tangent intersects. 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- The bisector of a chord passes through the center of the circle.

9.2 //**Definitions**// Inscribed Angle- An angle whose vertex lies on the circle and whose sides are chords of the circle.

Angle ACB is an inscribed angle

__//**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 the have the same measure.



heres another pic to understand wht all these line really mean



=9.3 Inscribed Angles and Arcs=


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

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 a a right angle. Arc-Intercept Corollary: If two inscribed angles intercept the same arc, then they have the same measure.

**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.

**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°

=9.5=

Tangent segment- the segment between the point of tangency and where its intersects with another line is know as the tangent segment Secant segment- the secant intersects the circle at 2 diffrent points, the secant segment is from the intersection point of the other line to the farthes point that intersects the circle. External secant segment- it is the part of the segment that does not enter the circle. internal secant segment- It is the segment inside the circle. Chord- a segment whos endpoints are on the line that is the outer circle.
 * Your job:**
 * //find and use special cases of segments related to circles, Include the following; secant-secant, secant-tangant, and chord-chord segments//
 * //make and use theorems about measures of the segments.//
 * Vocab:**

=Segments of Tangents, Secants, and Chords=
 * Things to know/Formulas:**
 * //**Things to know:** Radius=r, measure of the arc=M, lenght=L//
 * **//Formulas://** //L=M/360(2(PIE)r) and X= X/2 (case 1) (X1+X2)/2 (case 2) (X1-X2)/2 (case 3)//
 * Segments of Tangents, Secants, and Chords**
 * //**The Theorem that changed all Theorems:**// //If two you have two segments that are tangent to a circle from the same external point, then the two segments are the same.//
 * //**The Theorem of Math:** 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)//
 * //**The greater Theorem:** If a secant and a tangent intersect outside a circle, then the product of the lengths of the secant segment and its external secant segment equals the product of the tangent segment squared.(whole x outside =tangent squared)//
 * //**The last of the Theorems:** 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 chords//

//.//

9.6 Circles in the Coordinate Plane
Objectives: -Develop and use the equation of a circle. -Adjust the equation for a circle to move the center in a coordinate plane.

Definitions: There are no vocabulary terms within this section that haven't already been defined above.

Theorems: There are no theorems that needed to be known for the completion of this section.

Equations: Equation 1: "To derive the equation of a circle"- To find any point that is on a circle that's not on the x- or y- axis, you can draw a triangle whose legs have lengths of x and y. The length of the hypotenuse is the distance, //r//, from the point to the origin. To find the point, use this equation: //x squared + y squared equals r squared// Equation 2: "To move the center of a circle"- To find the standard form of the equation of a circle centered at a point not on the origin, study the diagram on the Holt Geometry book on page 612. The equation for that diagram is: //(x - h squared) + (y - k squared) = r squared//

Example: The center of this circle is at the origin (0,0). What are the intercepts of the circle? Solving: Look at the common intercepts of this section. Answer: X² + Y² has the center of (0,0).

SOME RElly swee links to help you visualize rhis stuff better

enjoy!

http://www.explorelearning.com/index.cfm?method=cResource.dspDetail&ResourceID=175