PETO626

=Chapter 9=

Section 9.1: Chords and Arcs.
__Objectives__ 1. Define a circle and its associated parts, and use them in constructions. 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.

A circle is the set of all points in a plane that are equidistant from a given point in the plane known as the center of the circle. A radius (plural, radii) is a segment from the center of the circle to a point on the circle. A chord is a segment whose endpoints line on a circle. A diameter is a chord that contains the center of a circle.
 * __Definitions__**
 * See diagram 9-1 for visual assistance.**


 * 9-1**

__**Definition: Central Angle and Intercepted Arc.**__ A central angle of a circle is an angle in the plane of a circle whose vertex is the center of the circle. An arc whose endpoints lie on the sides of the angle and whose other points lie in the interior of the angle is the intercepted arc of the central angle.
 * 9-2**

The degree measure of a minor arc is the measure of its central angle. The degree measure of a major arc is 360° minus the degree measure of its minor arc. The degree measure of a semicircle is 180°.
 * __Definition: Degree Measure of Arcs.__**

Example: The measure of a circle's central angle is 65°, find the measure of the circle's major arc. Answer: This may seem complicated, but if you think about it, it really isn't. Central angle=Minor arc, central angle=65°, so that means the minor arc must be 65°. The degree measure for a full circle is 360°, so to find the major arc it's 360-65, which is 295°, the answer.

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°(2(pi)r).

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

In a circle, or in congruent circles, the chords of congruent arcs are equal.
 * __Converse of the Chords and Arcs Theorem.__**

Section 9.2: Tangents to Circles.
1. Define tangents and secants of circles. 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.
 * __Objectives:__**

Secant: A line that intersects the circle at two points. Tangent: A line in the plane of the circle that intersects the circle at exactly 1 point. The 1 point is known as point of tangency.
 * __Definitions: Secants and Tangents.__**


 * 9-3**

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.
 * __Tangent Theorem:__**

A radius that is perpendicular to a chord of a circle cuts the chord in half.
 * __Radius and Chord Theorem:__**

If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is perpendicular to the circle.
 * __Converse of the Tangent Theorem:__**

The perpendicular bisector of a chord passes through the center of the circle.
 * __Theorem:__**

Section 9.3: Inscribed Angles and Arcs.
1. Define inscribed angle and intercepted arc. 2. Develop and use the Inscribed Angle Theorem and its corollaries.
 * Objectives:**

An angle whose vertex lies on a circle and whose sides are chords of the circle
 * __Definition: Inscribed Angle.__**

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

Example: I'm just coming up with this on the top of my head, but, say you want to make the angle measurements on a sports game, (it is a real job, trust me on that.) In a baseball game, the whole field is seen in a circle instead of the diamond were used to in baseball, a bat hits a ball at a certain angle, the angle is determined on where the ball hits the bat, and where the bat is when you make contact. Where the ball makes contact it makes a line going both ways and makes an arc intercepting the circle twice. If the inscribed angle is between 45° and 90° it's fair, if not it's foul. I truly hope that makes sense, it was really hard to explain.


 * __Definitions: Right Angle and Arc-Intercept Corollaries.__**
 * Right Angle Corollary:** If an 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.