Fian32

Objectives:

 * Define a circle and its associated perts,and use them in construction.
 * Define and use the degree measure of arcs.
 * Define and use the lenght measure of arcs.
 * Prove the theorem about chords and their intercepted arcs.

Definition: Circle
A circle is the set of allpoints in a plane that are equidistant from a given point in the plane known as the center of the cirlce. A **radius** is a segment from the center of a circle to a point on the circle. A **chord** is a segment whose endpoints line on a cirlce. A **diameter** is a chord that contains the center of a cirlce.

circle of arc is smaller than a semicirlce. about a fourth of a circle.
 * Minor Arc:**

Major Arc:
circle of arc is bigger than semicircle. about three fourths of a circle.

Definitions: Central Angle and Interceptig Arc
A **central angle** of a cirlce is an angle in the plane of a cirlce whose vertex is the center of the cirlce. An arc whose endpoints lie on the sides of the angle and whose other points lie in the intierior of the angle is the **intercepted arc** of the central angle.

Definition: Degree Measure of Arc
The degree of a minor arc 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

Arc Length:
If r is the radius of a cirlce 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(2pie//r//)

Objectives:
1) Define tangents and secants of circle. 2) Understand the relationship between tangents and certain radii of circles. 3) Understand the geometry of a radius perpendicular to a chord of a circle.

**Definitions:**
1) **Sectants** - A line that intersects a circle at two points. 2) **Tangents** - In a right triangle, the ratio of the lenght of the side opposite an acute angle to the length of the side adjacent to it. 3) **Point of tangency** - The point of a circle or sphere with a tangent line or plane.



Theorems:
1) **Tangent Theorem** - If a line is the tangent to circle, then the line is perpendicular to a radius of the circle drawn to the point of tangency. 2) **radius and chord theorem** - a radius that is perpendicular to a chord of a circle is bisect of the chord. 3) **Converse of the tangent theorem** - If a line is perpendicular to a radius of a circle at its endpoints on the circle, then the line is tangent to the circle 4) **Theorem** - The perpendicular bisector of a chord passes through the center of the circle.

Objections:
1) Define inscribed angle and intercepted arc 2) Develop and use the inscribed angle theorem and its corollaries.

Definitions:
1) **Inscribed angle** - is an angle whose vertex lies on a circle and whose sides are chords of the circle. 2) **Right-Angle Corollary-** If an inscribed angle intercepts a semicircle, then the angle is a right angle. 3) **Arc-Intercept-** If two inscribed angles intercept the same arc, then they have the same measure.

Theorems:
1)**Inscribed angle theorm:** The measure of a angle inscribed in a circle is equal to one-half the measure of the intercepted arc. 2)**Right Angle Corollary,** If an inscribed angle intercepts a semicircle, then the angle is a right angle. 3)**Arc-intercept Corollary,**If two insribed angles interrcept the same arc, then they have the same measure.

=9.4 Angles Formed by Secants and Tgangents= 1. Define angles formed by secants and tangents of circles. 2. Decelop and use theorems about mearsures of arcs intercepted by these
 * 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.
 * Theorem:**

The measure of an angle formed by two secants or chords that intersect in the interior of a circle is the of the measures of the arcs intercepted by the angle and its vertical angle.
 * Theorem:**

The measure of an angle formed by two secants that intersect in the exterior of a circle is the of the measures of the intercepted arcs.
 * Theorem:**

The measure of a secant - tangent angle with its vertex outside the circle is.
 * Theorem:**

The measure of a tangent - tangent angle with its vertex outside the circle is.
 * Theorem:**

=9.5 Segments of Tangents, Secants, and Chords.=

1.) Define special cases of segments related to circles, including secant-secant, secant-tangent, and cross-chord segment 2.) Develop and use theorems about measures of the segments.
 * Objectives**

- segment XA is a tangent segment - segment XB is a secant segment - segment XC is an external secant segment - segment BC is a chord __**Theorems:**__ If two segments are tangent to a circle from the same external point, then the segments are equal length. __**Theorem 9.5.2**__ If two secants intersect outside a circle, then the product of the lengths of one secant segment and external segment equals the product of the lengths of the other secant segment and its external segment. __**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. __**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.
 * __Theorem 9.5.1__**

=9.6 Circles in the Coordinate Plane=

Objectives:

 * 1)** Create an equation to graph a circle on the origin.
 * 2)** Create an equation to graph a circle not on the origin.

Equations:
X² + Y² = r ²
 * 1)** When the center of the circle is at the origin (0,0)

(X - h)² + (Y - k)² = r ² (h,k) is the origin.
 * 2)** When the center of the circle is not at the origin.

1)** A circle on the origin (0,0) and a radius of 5 x^2 + Y^2 = 5^2 (x - 5)^2 + (y - 4)^2 = 3^2
 * Example:
 * 2)** A circle with a center of (5,4) and a radius of 3