ScNa228

=__Chapter 9__=

9.1 Circles and Arcs
-Objectives-
 * 1) Define a circle and its associated parts, and use them in construction.
 * 2) Define and use the degree measure of arcs.
 * 3) Define and use the length measure of arcs
 * 4) Prove a theorem about chords and their intercepted arcs.

__**Vocabulary:**__
 * 1) -**Circle**: A set of points in a plane that are the same distance from the center of a circle.
 * 2) -**Chord**: A segment that extends from one edge point of a circle to another edge point of the same circle.
 * 3) -**Radius**: A segment that connects the edge of the circle to the center point of the circle.
 * 4) -**Diameter**: A segment that connects to two edge points of a circle making a 180° angle through the center point, or double the radius.
 * 5) -**Arc**: An unbroken part of a circle
 * 6) -**Semicircle**: Half of one circle.
 * 7) -**Minor Arc**: An arc that has a measure less than 180°
 * 8) -**Major Arc**: An arc that has a measure greater than 180°
 * 9) -**Central Angle**: An angle with a vertex at the center of a circle, with two end points on the edge of the circle.
 * 10) -**Intercepted Arc**: The arc of a central angle.

1.The measure of a minor arc is the measure of the central angle. 2.The measure of a major arc is 360° minus the degree measure of its minor arc. (A semicircle is 180°).
 * Degree Measure of Arcs**:


 * Arc Length**: M/360°(2(pi)r)


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


 * The Converse of the Chords and Arcs Theorem**: In a circle or in congruent circles, the chords of congruent arcs are

Label the following definitions on the circle below.** -What is the radius? -What is the chord? -What is the diameter? -Major arc? -Minor arc? (Image created by Nathan Schilling)
 * Example Question #1

9.2 Tangents and circles
-Objectives-
 * 1) Define tangents and secants of a 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.


 * __Vocabulary__:**
 * 1) **-Sectants:** A line that intersects the circle at two different points.
 * 2) **-Tangents:** A line that comes in contact with the circles edge.
 * 3) **-Point of tangency:** The point of a circle with a tangent line or plane.
 * 4) -**Theorem**: A chords perpendicular bisector passes through the middle of the circle.

According to the vocabulary above, which is the tangent and which is the secant?
 * Example Question #2**

(image created by Nathan Schilling)

If a line is a tangent to the circle then the line is perpendicular to the radius of the circle drawn to the point of tangency.
 * Tangent Theorem:**

A radius that is perpendicular to a chord of a circle bisects the chord. 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. [|Help]
 * Radius and Chord Theorem**:
 * Converse of a Tangent Theorem**:

9.3 Inscribed Angles and Arcs
-Objectives- 1. Define inscribed angles and intercepted arcs. 2. Develop and use the Inscribed Angle Theorem and its corollaries.

1. -**Inscribed Angle**: An angle that has a vertex that lies on the circles edge and the sides of the angle are the chords of the circle.
 * __Vocabulary__**:

The measure of an angle inscribed in a circle is equal to half the measure of the intercepted arc.
 * Inscribed Angle Theorem**:

An angle is a right-angle when an inscribed angle intercepts a semicircle.
 * Right-Angle Corollary**:

Two angles are the same measure if they have the same arc and they are inscribed.
 * Arc-Intercept Corollary**:

If the arc of the circle below is 89 then what is angle BAC? (Image created by Nathan Schilling)
 * Example Question #3**:

[|Help]

9.4 Angles Formed by Secants and Tangents
-Objectives-

1. Define angles formed by secants and tangents of circles. 2. Develop and use theorems about measures about measures of arcs intercepted by these angles.


 * Theorems:**

-If the vertex is on a circles edge then the angle is the arc divided in half. (X°/2) (x=arc) -If the vertex is outside the circles edge then the angle is each arc minus eachother divided by two. (X²-X¹)/2 -If the vertex is inside the circles edge then the angle is each added to one another divided by two. (X²+X¹)/2

Example Question #4: according to the info above how many degrees is angle BAC?

(Image created by Nathan Schilling) [|Help]

**9.5 Segments of Tangents, Secants, and Chords**
-Objectives-

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

-If two segments are tangent to a circle from the same external point, then the segments have an equal length. -If two secants intersect outside a circe, the product of the lengths of one secant segment and its external segment equals the outside segment and the whole segment. (whole X outside = whole X outside) If BC = 12, then what is DC? (Image created by Nathan Schilling) [|Help]
 * Theorems:**
 * Example Question #5:**

9.6 Circles in the Coordinate Plane
-Objectives- 1.Develop and use equation of a circle. 2.Adjust the equation for a circle to move the center in a coordinate plane.

Equations: 1.If the center of a circle lies on the origin the the equation is x² + y² = r². 2. If the center of a circle lies somewhere other than then the origin then the equation is (x-h)²+(y-k)²=r²

Create the correct equation using the following information. __Center__:(0,0) __Radius__:10
 * Example Question #6:**