rojo35

=CHAPTER 9=

9.1
=Chords and Arcs=
 * objectives:**
 * Define a circle and its associated parts, and use them in construction
 * Define and use the degree measure of arcs
 * Define and use the the lengh measure of arcs
 * Prove a theorem about chords and their intercepte arcs

Definitions:

 * CIRCLE- is the set of all points in a plane that are equidistant from a given point in a plane.
 * RADIUS- is a segment from the center of the circle to a point on the circle
 * CHORD- is a segement whose end points line on a circle.
 * DIAMETER- is a chord that contains the center of a circle
 * CENTRAL ANGLE- is an angle in the plane of a circle whose vertex is the center of the circle.
 * INTERCEPTED ARC- is an arc whose endpoints lie on the sides of the angle and whose other points lie in the interior of the angle
 * DEGREE MEASURE OF ARCS- the degree measure of a minor arc is the measure of its central angle. the degree measure of a major arc is 3360 degrees minus the degree of its minor arc.

Arc Length

 * L=M/360(2pie r)

Chords and Arc Theorem-
In a circle, or in congruent circles, the arcs of congruent chords are ? .

EXAMPLE-
If the minor arc is 70 then whats the major arc? you take the 360 minus the minor arc(70) 360-70= major arc. so your major arc is 290degrees

9.2
=Tangents to Circles= Objectives:
 * Define tangents and secants of circles
 * Understand the relationship between tangents and certain radii of circles
 * Understand the geometr of radius perpendicular to a chord of a circle

Secants and Tangents-
A secant to a circle is a line that intersects the circle at two points. A tangent is a line in the plane of the cirlce that intersects the circle at exactly one point, which is known as the point of tangency.

Tangent Theorem-
If a line is 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 the 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.

Theoreme-
The perpendicular bisector of a chord passes through the center of the circle.

EXAMPLE-
If the radius points (a,b) is 5 and if from the center to point x its 4 (a,x) then what is (a,b)? to find (a,b) you use the pythagorean theorem. so you take x^2+4^2=5^2 x^2+16=25 x^2=41 take square root of both sides answer is square root of 41

9.3
=[|Inscribed Angles and Arcs]= OBJECTIVES: Definitions: Inscribed angle- is an angle whose vertex lies on a circle and whose sides are chords of the circle.
 * Define inscribed angle and intercepted arc.
 * Develope and use the inscribed angle theorem and its corollaries

Inscribed Angle Theorem-
The measure of an angle inscribed in a circle is equal to ? the measure of the intercepted arc.

Right Angl Corollary-
If an inscribed 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.

EXAMPLE-
if angle ABC is inscribed to circle X and intercepts arc AC. AC=45degrees then what is the angle? 1/2 (arc AC) 1/2(45) answer 22.5

9.4
=[|Angles Formed by Secants and Tangents]=
 * OBJECTIVES:**
 * Define angles formed by secants and tangents of circles.
 * Develope and use theorems about measures of arcs intercepted by these angles

Theorem-
If a tangent and a secant (or a chord) intersects on a circle at the point of tangecy, then the measure of the angle formed is ? the measure of its intercepted arc.

Theorem-
The measure of an angle formed by two secants or chords that intersect in the interior of a circle is ? the ? of the measures of the arcs intercepted by the angle and its vertical angle

Theorem-
The measure of an angle formed by two secants that intersect in the exterior of a circle is ? the ? of the measures of the intercepted arcs.

EXAMPLES-
In theorem 1 what would you do if arc AC was 80 and you needed to find the triangle? 1/2 of arc length AC answer 40 In theorem 2 what would you do if two secants meet inside the circles and arc AC DE were 40 and 80 what is the angle?

you take (1/2) DE+AC (1/2) 80+40 =60 In theorem 3 what do you do if there are to secants that intersect outside the cirlce and the arc lenghts are 80 and 20? you take (1/2) 80-20=x (1/2)60 =30

9.5
=[|Segments of Tangents, Secants, and Chords]=
 * OBJECTIVES:**
 * Define special cases of segments related to circles, including secant-secant, secant-tangent, and chord-chord segments
 * Develop and use theorems about measures of the segments.

Theorem-
If two segments are tangent to a circle from the same external point, then the segments ? .

Theorem-
If two secants intersect outside a circle, the prodct of the lengths of one secant segment and its external segment equals ? . (Whole x Outside = Whole x Outside)

Theorem-
If a secant and a tangent intersect outside a circle, then the product o the lenghts of the secant segment and its external segment equals ? . (Whole x Outside = Tangent Squared)

Theorem-
If two chords intersect inside a circle, then the product of the lenghs of the segments of one chord equals ? .

EXAMPLE-
If two tangents are tangent to the same circle from the same point if tangent AX is 42 then whats BY? 42 because they are equal

9.6
=[|Circles in the Coordinate Plane]= = = OBJECTIVES:
 * Develop and use the equation of a circle
 * Adjust the equation for a circle to move the center in a coordinate plane.

EXAMPLE-
Given x^2+y^2=36 your coordinates would be (0,0) the radius would be (6) because you take the square root of 36

Given (x-3)^2+(y-8)^2=144 your coordinates are the opposite of the x and y so it would be (3,8) the radius would be square root of 144 so (12)