JOURDAN!

=__Chapter 9__=
 * 9.1 - Chords and Arcs
 * 9.2 - Tangents to Circles
 * 9.3 - Inscribed Angles and Arcs
 * 9.4 - Angles Formed by Secents and Tangents
 * 9.5 - Segements of Tanents, Secants, and Chords
 * 9.6 - Circles in the Coordinate Plane

=**__Chapter 9.1__**=

The Objectives in this section are as follows:
 * Define a circle and its associated parts, and use them in a construction.
 * Define and use the degree meause of arcs.
 * Define and ise the length measure of arcs.
 * Prove a theorem about chords and their intercepted arcs.
 * __Definition of a **CIRCLE:**__
 * a circle is the set of all points in a plane at a fixed distance called the **radius**. The circles out side is called a **circumference,** the distance all the way around, the length of the circle. To explain more that mean that you can set one point on the outer edge of the circle and a multiple number of points everywhere on the circle, and no matter what the distance for any point to the center will be equal. Also in a circle there can be lines that are called **chords**.
 * 1) **__Radius:__** A segment from the center of the circle to any point on the circle. That distannce fromt he center to a outer point is called a __RADIUS.[[image:Picture_8.png]]__
 * 2) **__Circumference:__** The distance around a circle; the formula for circumference of a circle is pi times the diameter (C = TTd).[[image:Picture_7.png]]
 * 3) **__Chord:__** A line segment joining two points on a circle.


 * __Major and Minor Arcs:__**
 * __**Arc:**__ is a continuous portion of a circle. There are 3 different kinds of ARCS.
 * 1) **__Semicircle:__** A semicircle is point picked points that forms half of a circle. Being half of a circle's 360°, the arc of a semicircle will always measure 180°. You name the semicircle arc by the two endpoints plus a point in between. Say the endpoints are A and C and the point B is in between them. The arc would be named ABC.[| SEMICIRCLE]
 * 2) __**Minor Arc:**__ is the arc of a circle that is less than the length of the the semicircle. Going back to the last example for naming a arc. This arc would be named AB or BC because it is less then the ABC.
 * 3) __**Major Arc:**__ arc of a circle that is greater than the semicircle. MORE THAN 180°. That would also be named as a semicircle arc would be named, endpoints and another point that lies on the MAJOR ARC.

the blue is a MAJOR ARC and the red is a MINOR ARC.

__Definiton of **Central Angles** and **Intercepted Arcs:**__ __Definiton of **Degree Measure of Arcs:**__
 * __**Central Angles: A central angle is an angle whose vertex is the center of a circle, and whose sides pass through a pair of points on the circle.**__
 * __**Intercepted Arc:**__ is a arc whose endpoints lie on the sides of the angle and whose other points lie in the interior of the angle. That is the intercepted arc of the centrel angle. The arc of a circle within an inscribed angle.
 * **__Degree Measure for a MINOR ARC:__** the measure of its central angle**__.__**
 * **__Degree Measure for a MAJOR ARC:__** is 360° minus the degree measure of its minor arc.
 * __**Degree Measure for a semicircle:**__ is just 180°

r is the radius. M is the degree measure of an arc of the circle. L is the length. To find out the arc length you use this equation...
 * __Arc Length:__**
 * L = M ÷ 360° (2TTr)

In a circle or in congruent circles, the arcs of congruent chords are congruent.
 * __Chords and Arcs Theorem:__**

Example: Determine the degree measure of an arc with the given length, L, in a circle with the given radius, r. 1. L= 10, r = 50 HOW TO FIGURE IT OUT: 10 = m (2TT50) 360 divide everything by (2TT50) and that just crosses itself out 360 (1÷20TT) = (m/360) 360 360 and 360 cross each other out = 226.19

=__Chapter 9.2__=

The Objectives in this section are as follows:
 * Define tangents and secants or circles
 * Understand the relationship between tangents and certain radii or circles. (radii = plural form of radius)
 * Understand the geometry of a radius perpendicular to a chord of a circle.

__Defintion of a **Secent and Tangent**:__
 * 1) **__Secant:__** A line that crosses the circle only twice. This line runs right through a circle. It is not a diameter, the length of a secant is smaller, and does not run perfectally through the center of a circle.
 * 2) **Tangent:** a line intersecting only one point of a circle also called point of tangency.


 * __THEOREMS:__**
 * **__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:__** Radius and Chord Theorem:Radius and Chord Theorem: A radius that is perpendicular to a chord a circle biscets the chord.
 * **__Converse of the Tangent Theorem:__** Converse of the Tangent Theorem:Converse of the Tangent Theorem: If a line is a perpendicular to a radius of a circle at its endpoints on the circle, the the line is tangent to the circle.

[|WHAT ONE IS A TANGENT AND A SECANT ON A CIRCLE?]

=__Chapter 9.3__=

The Objectives in this section are as follows:
 * Define inscribed angles and intercepted arc.
 * Develop and use the inscribed angle theorem and its corollaries.

[|INSCRIBED ANGLE]
 * __Inscribed angle:__** An angle placed inside a circle with its vertex on the circle and whose sides contain chords of the circle.
 * **__Inscribed Angle Theorem:__** The measure of an angle inscribed in a circle is equal to the measure of the intercepted arc.

[|Right-Angle Corollary]
 * __Right-Angle Corollary:__** It is when a inscribed anglw intercpets a semicircle, then the angle is a right angle.

__**Arc-Intercept Corollary:**__ It is when two inscribed angles intercept the same arc, the they have the same measure.

Example 1:

Arc AC = 50° Angle ABC is insribed in POINT O and intercepts ARC AC, by the Inscribed Angle Theorem. m ANGLE ABC =1/2 m ARC AC= 1/2 (50°) = 25°

=__Chapter 9.4__=

The Objectives in this section are as follows:
 * Define angles formed by secants and tangents of circles.
 * Develop and use theorems about measures of arcs intercepted by these angles.

There are three cases to consider when you are looking at angles formed by pairs of lines that intersect a circle in two or more places. __**Case 1:** Vertex is on the circle.__ 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.
 * Secant and a Tangent
 * Two Tangents
 * __Theorem:__**

__**Equation:**__ (x/2)°

__**Case 2:** Vertex is inside the circle.__ The measure of an angle formed by two secants of chords that intersect in the interior of a circle is half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
 * Two Secants
 * __Theorem:__**

__**Equation:**__ X1+X2 ÷ 2

__**Case 3**: Vertex is outside the circle.__ __**Theorem:**__ The measure of an angle formed by two secants that intersect in the exterior of a circle is half the of the measures of the intercepted arcs.
 * Two Tangents
 * Two Secants
 * Secant and a Tangent

__**Equation:**__ X1-X2 ÷ 2

= = =__Chapter 9.5__=

The Objectives in this section are as follows: [|Which one is a Tangent, a Secant, a External Secant, and a Chord?]
 * 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.

__Segments formed by Tangents:__
 * __Theorems:__**
 * If two segments tangents to a circle from the sane external point, then the segments are equal.

__Segements formed by Secants:__
 * If two secants out a circle, the product of the length of one secant segment and its external segment equals the product of the length of one secant segment and its external equals the product.
 * (Whole x Outside = Whole x Outside)
 * If a secant and a tangent intersect a circle, then the product of the lengths of the secant segment and its external segment equals the product of the length of the secant segment and its external segment equals the product.
 * (Whole x Outside = Tangent Squared)

__Segments formed by Interesting Chords:__
 * 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 one chord.

=__Chapter 9.6__=

The Objectives in this section are as follows:
 * Develop and use the equation of a circle.
 * Adjust the equation for a circle to move the center in a coordinate plane.

Example: 1. Equation for a circle with a center at the origin is, x² + y² = r². r is the radius find the x and y intercepts

The intercepts are: (9,0), (-9,0) , (0,9) , (0,-9)

The equation is: x² + y² = 3²