SHAN,23

__CHAPTER 9 WIKI NOTES__
9.1 - Chords And Arcs 9.2 - Tangents And Circles 9.3 - Incribed Angles And Arcs 9.4 - Angles Formed By Secants And Tangents 9.5 - Segments Of Tangents, Secants, And Chords 9.6 - Circles In The coordinate Plane

__9.1 Chords And Arcs__
Circle - A set of all points in a plane that are equidistant from a given point known as the center of the circle Radius - A segment from the center of the circle to a point on the circle Chord - A segment whose endpoints line on a circle Diameter - A chord that contains the center of the circle Arc - A unbroken piece of the outside of a circle Semicircle - An arc whose endpoints are the endpoints of a diameter Minor Arc - An arc that is shorter then a semicircle or 180 degrees Major Arc - An arc that is longer then a semicircle or bigger then 180 degrees Central Angle - An angle in the plane of a circle whose vertex is the center of the circle Intercepted Arc - An arc whose endpoints lie on the sides of the angle and whose other points lie in the interior of the angle From Wikispaces
 * 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

Find the length of the indicated circle. The radius equals 170 The length of the arc is 1/20 of the circumference of the circle. Remember that C=2pir Length of arc=1/20(2piX17017pi53.4=53Arc 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(2pir) Chords and Arc Theorem- In a circle, or in conguent circles, the arcs of congruent chords are similar. The Converse of the Chords and Arcs Theorem- In a circle or in congruent circles, the chords of congruent arcs are similar.



__9.2 Tangents To Circles__
From Wikispaces
 * 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

Secant- A line of a circle that intersects the circle at two points Tangent- A line in a plane of the circle that intersects the circle at exactly one point Point Of Tangency- Point at which the tangent touches the circle it is on

Three possibilities of where the tangents or secants may intersect the circle: 2 points of intersection, 1 point of intersection, 0 points of intersection

Tangent Theorem: 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. Radius and Chord Theorem: A radius that is perpendicular to a chord of a circle bisects the chord. Converse of the Tangent Theorem: If a line is perpendicular to a radius of a circle at its endpoints on the circle, then the line is tangent to the circle. Theorem: The perpendicular bisector of a chord passes through the center of the circle.

Circle P has a radius of 5 in. and PX is 3in. Line PR is perpendicular to line AB at point X. Find AB

By the Pythagorean Theorem: (AX)x(AX) + 3x3 = 5x5 (AX)x(AX) = 5x5 - 3x3 (AX)x(AX) = 16 AX = 4 By the Radius and Chord Theorem, line PR bisects line AB, so BX AX 4. Therefore, AB = AB + BX 4 + 48

__9.3 Inscribed Angles and Arcs__
From Wikispaces
 * Define inscribed angle and intercepted arc
 * Develop and use the Inscribed Angle Theorem and its corollaries

Inscribed Angle- An angle whose vertex lies on a circle and whose sides are chords of the circle

Inscribed Angle Theorem: The measure of an angle inscirbed in a circle is equal to half the measure of the intercepted arc. Right-Angle 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.

Find the measure of <XVY <XVY is inscribed in circle P and intercepts arc XY. By the Inscribed Angle Theorem: m<XVy 1/2marc XY 1/2(45) = 22 1/2

**__9.4 Angles Formed by Secants and Tangents__**
From Wikispaces
 * Define angles formed by secants and tangents of circles
 * Develop and use theorems about measures of arcs intercepted by these angles

CASE 1: Vertex is on the circle A) Secant and Tangent B) Two Secants CASE 2: Vertex is inside the circle A) Two circles CASE 3: Vertex is outside the circle A) Two Tangents B) Two Secants C) Secant and Tangent

Theorem: If a tangent and a secant (or a chord) intersect on a circle at one point of tangency, then the measure of the angle formed is half the measure of its intercepted arc. Theorem: The memasure of an angle formed by two secants or chords that intersect in thee interior of a circle is half the length of the measures of the arcs intercepted by the angle and its vertical angle. Theorem: The measure of an angle formaed by two secants that intersect in the exterior of a circle is half the length of the measures of the intercepts arcs.

__9.5 Segments of Tangents, Secants, and Chords__
From Wikispaces
 * 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 intercept. Theorem: If two secants intersect outside a circle, the product of the lengths of one secant and its external equals (whole x outside = whole x outside) Theorem: If a secant and a tengent intersect outside a circle, then the product of the lengths of the secant segment and its external segment equals (whole x outside = tangent squared)

EX= 1.31, GX= 0.45, FX= 1.46, Find HX (W x O = W x O) EX x GX = FX x HX 1.31 x 0.45 = 1.46 x HX 1.46 x HX = 0.5895 HX = 0.40

The distance from the station to the lake is 300 yrds along the road tangent to the lake, and 50 yrds along a straight line to the nest. How far is the nest from the ranger station? The road is a tangent and the line to the nest is a secant. They intersect outside the circle. D is the distance from the station to the nest (W x O = Tangent Squared) D x 50 = 300 x 300 D x 50 = 90, 000 D = 1800 yrds, roughly 1 mile ( 1 mile 1760 yrds )

__9.6 Circles in the Coordinate Plane__
From Wikispaces
 * Develop and use the equation of a circle
 * Adjust the equation for a circle to move the center in a coordinate plane

Given: X x X + Y x Y = 25 To find the x-intercepts, find the values of X when Y=0. X x X + 0 = 25 X x X = 25 X = +- 5