kiin16

=Chapter 9 Circles=



Objectives:

 * Define a circle and its associated parts, and use them in constructions.
 * Define and use the degree measure of arcs.
 * Define and use the length measure of arcs.
 * Prove a theorem about chords and their intercepted arcs.



Definitions:
- Circle: A plane containing a set of points that are all an equal distance from the center of the circle. - Radius: A line segment that goes from the center of the circle to a point on the edge of the circle. - Chord: A line segment whose endpoints meet at a circle. - Diameter: A chord where the center of the circle is in the middle, between the two endpoints. - Arc: Any part of a circle that is unbroken. - Endpoints: The points on or in a circle in which a line segment ends. - Semicircle: When the endpoints of an arc are the same as the endpoints of the diameter. - Minor Arc: An arc that is shorter than the semicircle of the circle in which it is in. - Major Arc: An arc that is longer that the semicircle of the circle in which it is in. - Central Angle: An angle whose vertex is the center of the circle that it is in. - Intercepted Arc: When the endpoints of an arc are on the sides of the central angle and the other points are inside the angle.

Degree Measure of Arcs:
~ Minor Arc: The degree measure of this arc is the measure of its central angle. ~ Major Arc: The degree measure of this arc is 360° minus the degree measure of its minor arc. ~ Semicircle: The degree measure is always 180°.

Arc Length:
L=M/360°{2(3.14)r}

Here is a website with an activity to help you understand and practice. http://www.explorelearning.com/index.cfm?method=cResource.dspDetail&ResourceID=175

Example Questions:
How do you define a circle? A radius? A major and minor arc? A chord? And a central angle?

Objectives:

 * Define tangents and secants of circles
 * Understand the relationship between tangents and certain radii of circles
 * Understand the geometry of a radius perpendicular to a chord of a circle

Definitions:

 * Secant: A line segment of a circle that intersects the circle at two points
 * Tangent: A line segment on the intersecting the circle on its outside line in exactly one point, called the --->
 * ---> Point of Tangency.

A line is perpendicular to a circle's radius that is drawn to the point of tangency if the line is tangent.

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

Converse of Tangent Theorem:
A line is tangent to a circle if that line is perpendicular to the circle's raduis at its one endpoint falling on the circle's edge.

This website further explains how a tangent works, and displays what a tangent looks like. http://library.thinkquest.org/10030/13tangen.htm

Example:
Question: What do you use to find the measure of a chord if you know the measures of the chord perpendicular to it? Answer: The Pythagorean Theorem.

Objectives:

 * Define intercepted arc and inscribed angle
 * Use the Inscribed Angle Theorem and its corollaries.

Inscribed Angle Theorem:
An angle is inscribed if its sides are chords of the circle whose vertex it lies on.

Right-Angle Corollary:
An inscribed angle is a right angle if it intercepts a semicircle.

Arc-Intercept Corollary:
If an arc is intercepted by two inscribed angles, those angles have the same measures.

This website not only will help you better understand these theorems, but it also has a way to apply them. http://www.ies.co.jp/math/java/geo/enshukaku/enshukaku.html

Example:
The insrcibed angle of circle P is angle LMN. The chord that it intersects is measured at 90°. Find the measure of LMN. Solution: You have to divide the chord into two. 90 divided by 2 equals 45. Therefore, angle LMN equals 45°.

Objectives:

 * Learn which angles are formed by tangents and secants.
 * Use theorems that state how measures of arcs are intercepted by certain angles.

Theorem One:
If a tangent and a secant or chord intersect on a circle at the point of tangency, then the measure of the angle's intercepted arc is twice the measure of the angel formed.

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

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

This website has a listing of all the possible angle combinations. http://regentsprep.org/regents/mathb/5A1/CircleAngles.htm

Example:
There is a circle that is intersected by a tangent and a secant, which forms the angle ABC. The minor chord that it intersects is 140°. Find the measure of angle ABC. Solution: To find the measure of ABC you divide the chord it intersects by 2. So 140/2=70. ABC is 70°.

Objectives:

 * Define circles whose segments have special cases. Some of these include secant-tangent, secant-secant, and chord-chord segments
 * Put to use theorems defining measures of segments.

Theorem One:
Two segments are equal if they are tangent to a circle from the same external point.

Theorem Two:
(whole x outside=whole x outside)

Theorem Three:
(whole x outside=tangent squared)

Example:
There are two secants that intersect at point E, AC and BD. AC=3, AE=6, and DE=4. Find BD. Solution: (whole x outside=whole x outside) 6 x 3=(4+x)4 18=16+x x=2

The x- and y-intercepts
To find the y-intercept, replace the x-intercept with 0. For Example: x² + y²=49 0 + y²=49 y=7 To find the x-intercept, just replace y with 0.

Writing Equations
If given the center and radius of a circle, you can write the equation of the circle. For Example: center:(0,0); radius:3 x²+y²=6

To find the center and radius of a circle, go backwards. Use the given equation. For Example: x²+y²=64 center:(0,0); radius:8