dush218

=Chapter 9=

Main objectives:

 * Define a circle and the different part and use them.
 * Define and use the length and degree measure of arcs.
 * Prove the Chords and Arc Theorem.

Vocabulary:

 * **Circle:** A set of points in a plane that are an equal distant from the center of the circle.
 * **Radius:** A segment that is connects the center of the circle to a point on the outside of the circle.
 * **Chord:** A segment that extends from one point on the circles edge to another point on the circles edge.
 * **D****iameter:** A segment that splits the circle in half by going through the center of the circle.
 * **Semicircle:** half of a circle.
 * **Minor Arc:** An arc that measures smaller than 180°(or a semicircle).
 * **Major Arc:** An arc that measures greater than 180°(or a semicircle).
 * **Central Angle:** An angle with a vertex at the center of the circle and two endpoints on the edge of the circle.
 * **Intercepted Arc:** The arc of a central angle.

How to Measure an arc:

 * In a //**minor arc**// it is the measure of the central angle.
 * In a //**major arc**// you subtract the measure of the minor arc by 360°.
 * In a //**semicircle**// it is 180°.

How to Find the Length of an arc:

 * **r** is the radius
 * **M** is the measure of the circles arc (in degrees).
 * **L** is the length
 * L=M/360°(2****πr)**

Chords and Arcs Theorems:
In circles the arcs of the congruent chords are congruent.

The Converse Of the Chords and Arcs Theorem:
In circles the chords of congruent arcs are congruent.

Example 1:
(scd Answers Here scd)
 * 1) Name the Radius of the orange
 * 2) Name the Diameter
 * 3) Name a chord
 * 4) Name the arc. Is it major or minor?

__Need m[|ore help? Click here]__

Main Objectives:

 * Define the tangents and secants of a circle.
 * Understand the relationship between the radii and the tangents in a circle.
 * Understand how a radius is perpendicular to a chord of a circle.

Vocabulary:

 * **Secant:** A line that goes through the circle and intersects at two points.
 * **Tangent:** A line that goes through the circle and intersects at one point.
 * **Point of Tangency:** The point at which a tangent intersects.

Tangent Theorem:
When 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.

Radius and Chord Theorem:
When a radius is perpendicular to a chord of a circle bisects the chord.

Converse Of Tangent Theorem:
When a line and a radius are perpendicular then the line is a tangent to the circle.

Chord and Circle Theorem:
The perpendicular bisector of the chord passes through the center of the circle.

Example 2:
If the radius is 7 units. AE is 4 units. AC is 9 units. AB is perpendicular to CD at point E. Find CD. (scd Answer Here scd)

[|Need more help? Click here.]

Main Objectives:

 * Define inscribed angles and arcs.
 * Understand and use the inscribed angles theorem.

Vocabulary:

 * **Inscribed angle**: When the angles vertex is on the edge of a circle and the sides of the angle are chords in the circle.

Inscribed Angle Theorem:
The inscribed is measured half as much as the intercepted arc. (created by Shelby)

Right-Angle Corollary:
When an inscribed angle is inside a semi-circle the angle is a 90°(right angle).

Arc-Intercept Corollary:
When 2 inscribed angles are intercepting the same arc, it makes them the same measure. (created by Shelby)

Example 3:
Arc BC is 78° Find The measure of angle BAC. (scd Answer Here scd)

__Need m[|ore help click here]__ [|Or here] [|Or here..]

Objectives:

 * Define angles formed by tangents and secants.
 * Develop and use some theorems about measures of arcs intercepted by these angles.

Theorems:
When two lines intersect **inside** the circle the measure of the intercepted angle is half the sum of the measure of the arcs intercepted by the angle and its vertical angle[ (X1°+X2°)/2.
 * When two lines intersect **on** the circle the measure of the intercepted angle is half the measure of the intercepted arc (X°/2).
 * When two lines intersect **outside** the circle the measure of the intercepted angle is half the difference of the measure of the arcs intercepted by the angle and its vertical angle[ (X1°-X2°)/2.

Example 4:
In circle C find arc BED
 * 1) If angle BDA is 120°
 * 2) If angle BDA is 89°

(created by Shelby)

(scd Answer Here scd)

__Need m[|ore help? Click here]__

Objectives:

 * Define special cases of segments related to circles.
 * Develop and use theorems about the cases of segments related to circles.

Theorems:

 * If there are two tangents from the same external point, then the segments have an equal length.
 * If two secants intersected outside a circle, then the product of the whole segment and the outside segment equals the product of the whole segment and the outside segment.(Whole × outside=Whole × outside)
 * If two chords intersect inside the circle, then the product of the whole segment and the outside segment equals the length of tangent segment squared. ( Whole × outside=Tangent squared)
 * If two chords intersect inside a circle, then the product of part one and part two on one chord equal the product of part one and part two of the other chord. (Part 1 × Part 2 = Part 1 × Part 2)

Example 5:
In circle Z below, segment //XW// and segment //XY// is a tangent at points //W// and //Y//.

(created by Shelby)
 * 1) If //WX//=12, then //XY//= ?
 * 2) If //ZW//=6 and //XY//= 8, then //ZX// = ?

(scd Answer Here scd)

__Need m[|ore help? Click here]__

Objectives:

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

Equations:

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


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

> ===Example 6:=== > (scd Answer Here scd) > > __Need m[|ore help? Click here]__ > > > > > >
 * 1) What is the equation if the center is (4,-7) and the radius is 6?
 * 2) Graph a circle with a center of (0,0) and a radius of 5. What is the equation?