scab518&nbsp;


 * Chapter 9

Chapter 9 section 1

//Objectives://**
 * //**Define a circle and its associated parts, and us 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: set of all points in a plane that are equidistant. Radius: is a segment from the center of the circle ot a point on the circle. Chord: is a segmetn whose endpoints line a circle. Diameter: a diameter is a chord that contains the center of a circle. Arc: unbroken part of a circle. Endpoints: is the distinct parts on a circle divided by two arcs. Semi-circle: are whose endpoints are endpoints to a diameter. Minor arc: of a circle is an arc that is shorter than a semi-circle. Major arc: is a cirlce that is an arc that is longer than a semi-circle of that circle. Central angle: cirlce is an angle in a plane of a circle whose vertex is the center of the circle. Intercepted arc: endpoints on the sides of an angle that point to the interior of the angles. Arc length: R is the radius of a circle and M is the degree of measure of a arc the circle, then the length of the arc given by the following: L=M/360 times ( 2 times pi times R)**//





to show what radius, chord and diameter are.

The degree measure of a minor arc is the measure of its central angle. The degree measure is 360 degrees minus the degree of its minor arc.**
 * Degree Measure of Arcs



Example:

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

Chapter 9 section 2**


 * //Objectives://**
 * //* Tangents and Secants of circle.//**
 * //* Understand the relationship between tangents and certain radii of circles.//**
 * //* Understand the geometry of a radius perpendicular to a chord of a circle.//**

Secant: circle is a line plane of the circle that intersects the circle exactly two point. Tangent: line in the plane of the circle that intersects the circle a one point. Point of Tangency: Is a line that intersects a circle at one point.//
 * //Definitions:



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



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



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



Chapter 9 section 3

//Objectives://**
 * //**Define inscribed angle and intercepted arc.**//
 * //**Develope and use the inscribed Angle theorem and its corollary.**//

//**Definitions: Inscribed Angle: Is a angle thats vertex lies on the circle and whose sides are the chords of the circle.

To find the inscribed angle you must first know the arc measure. To find the inscribed angle you divide the arc measure in half. So say that the arc length was 120 the inscribed angle would then be 60 degrees.

Theorems: Inscribed angle theorem: the measure of an angle inscribed in a circle is equal to half the measure of the intercepted arc.

Corollaries: Right-angle Corollary**:// **//if an inscribed angle intercepts a semicircle, then the angle is a right angle.

Arc-intercept corollary: if two inscirbed angles inercept the same arc, then they have the same measure.//



Chapter 9 section 4

Objectives:**
 * **define angels formed by secants and tangents of circles.**
 * **develope and use theorems about measures of arcs intercepted by these angles.**


 * //Cases:

Case 1: Vertex is on the circle.//**

Vertex is inside the circle.//**
 * //Case 2:



Vertex is outside the circle.//**
 * //Case 3:



If a tangent and a secant ( or a chord) intersects on a circle at the point of tangency, then the measure of the angle formed is//** **//__the measure of its intercepted arc.
 * //Theorem 1 :

Theorem 2:__ The measure of an angle formed by two secants or chords that intersect in the interior of a circle is half the sum of the measures of the arcs intercepted by the angle and its vertical angle.



__Theorem 3:__ The measure of an angle formed by two secants that intersect in the exterior of a circle is half the sum of the measures of the intercepted arcs.



Theorem 4: The measure of a secant-tangent angel with its vertex outside the circle is_//****//__.

Theorem 5: The measure ot a tangent-tangent with its vertex outside the circle is__.//

Chapter 9 section 5 //Objectives://**
 * **//Define special cases of segments related to circles, including secant-secant, secant-tangent, and chord-chord segments.//**
 * **//Develope and use theorems about measures of the segments.//**


 * //Theorems:

Theorem 1: If two secants are tangent to a circle from the same external point, then the segments//**__**//.

Theorem 2: If two secants intersect outside a circle, the product of the lengths of one secant segment and its external segments equals//**__**//__. ( whole x outside = whole x outside )

Theorem 3: If a secant and a tangent intersect outside a cirlce, then the product of the lengths of that secant segment and its external segment equals___. ( whole x outside = tangent squared )

Theorem 4: If two chords that intesect on the inside of a circle, then the product of the lengths of the segments of that chord equals_.//**

//**Objectives:**// Are the points where the circle comes in to contact of the X or Y lines.//** Example of this would be: X^2 + Y^2 = 41 so X^2 + Y^2 = 41 X^2 = 16 X = 4 Y^2 = 25 Y = 5
 * Chapter 9 section 6**
 * **//Develope and use the equation of a circle.//**
 * **//Adjust the equation of a circle to move the center in a coordinate plane.//**
 * //Points of interception:
 * //This section we will be working with X squared and Y squared equals radius squared.

Moving the center of a circle: You take a center points G and L for example and use the equation ( X - H )^2 + ( Y - K )^2 = R^2. Or you can use the equation X^2 + Y^2 = R^2.//**

( X + 4)^2 + ( Y + 2)^2 = 4 Is the equation that the graph of the circle represents. Now graph your own circle with this equation: ( X + 4 )^2 + ( Y + 6) ^2 = 49. What is the radius and the points of interception.

X^2 + Y^2 = 4 This is an equation that is represtented in the graph above. Now graph your own circle with the equation: X^2 + Y^2 = 36. Find the radius and the points of interception.

Answers to examples in pictures: 1. was to find the arc length of AB 1b. the answer should be 17.16

2. was 190 for the arc AV 2b. answer should be 95 degrees

3. was 90 degrees for GH and 70 for EF 3b. the answer should be 75 degrees

4. was to put 120 for mRS and 30 for mUT 4b. you should have 45 degrees as a answer.

5. ( X + 4 )^2 + ( Y + 6) ^2 = 49 find the radius and points of interception 5b. the radius should be 7 and the points of interception should be 6. X^2 + Y^2 = 3 find the radius and points of intercept 6b. radius is 6 and points of intercept are (0,6) (0, -6) (6,0) (-6, 0)