Erin's+Chapter+9

Chords and Arcs**
 * Section 9.1

Objective: Define and use the degree measure of arcs. Objective: Define a circle and it's assosciated parts, and use them in constructions. Objective: Define and use length measure of arcs Objective: Prove a theorem about chords and their intercepted arcs.

__Definitions:__ Circle: Set of all points in a plane that are equidistant from a given point in the plane: center of the circle. Radius: Segment from the center of the circle to a point on the circle. Chord: A segment whise endpoints line on a circle. Diameter: A chord that contains the center of a circle. Arc: An unbroken part of a circle. Endpoints: Any two distinct points on a circle that divide the circle into two arcs. Semi-Circle: an arc whose endpoints are endpoints of a diameter. Minor Arc: An arc that is shorter than a sami-circle of that circle. Major Arc:An arc that is longer than a semi-circle of that circle. 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 angle and whose other points lie in the interior of the angle.

__Degree Measure of Arcs:__ Minor: the measure of its central angle Major: 360 degrees minus the degree measure of its minor arc. Degree measure of a Semi-Circle: 180 degrees.

__Example One:__

Find the measures of arcs RT, TS and RTS.

The measures of arc RT and arc TS are found from their central angles.

arc MRT 100 degrees. arc MTS 90 degrees

arc RT and TS are adjacent angles. add their measures together to find the measure of arc RTS.

measure of arc RTS = measure of arc MRT+ MTS =100 degrees + 90 degrees. 190 Degrees.

__Example Two:__ Find length of the arc

r =170mm length= 1/20 of the circumference of the circle. C=2 pi r Length of arc: 1/20(2pi x 170) =17pi, approx. 53 mm

Arc 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 (2pi x r)

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

Tangents to Circles**
 * Section 9.2

Objective: Define tangents and secants of circles. Objective: Understand the relationship beterrn tangents and certain radii of circles.

__Defenitions:__ Secent: a line that intersects the circle at two points. Tangent: A line in the plane of the circle that intersects the circle at exactly one point. Point of Tangency: The one point where the tangent intersects the circle.

__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.

__Example:__

cirlce P has a radius of 5 in. and px is 3 in. PR is perpendicular to AB at point x. Find AB.

(Ax)^2+3^2=5^2 (Ax)^2=5^2-3^2 (Ax)^2=16

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

Inscribed Angles and Arcs**
 * 9.3

__Objectives:__ 1. Define inscribed angle and intercepted arc 2. Develop and use the Inscribed Angle Theorem and its corollaries.

__Definitions:__ Inscribed angle- is an angle whose vertex lies on a circle and whose sides are chords of the circle.

__Example One__ Find the measure of XVY if the outer arc is 45 degrees, and if XVY is inscribed in P and intercepts XY.

__Solution:__ Inscribed Angle Theorem: m XVY 1/2(45) 221/2

__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.

Angles Formed by Secants and Tangents**
 * Section 9.4

__Objectives:__ 1. Define angles formed by secants and tangents of circle. 2. Develop and use theorems about measure of arcs intercepted by these angles.

__Theorem-__ If a tangent and a secant or a chord, intersect on a circle at the point of tangency then the measure of the angle formed is one-half the measure of its intercepted arc.

__Theorem-__ The measure of and angle formed by two secants or chords that intersect in the interior of a circle is one-half the sum of the measure 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 one-half the difference of the measure of the intercepted arcs.

__Example one:__ If segment C is tangent to a circle and AV is a secant to the circle, the two intersect on the circle, and the exterior arc is 150 degrees, what is the measure of angle AVC?

__Solution:__ measure of angle AVC = 1/2 the measure of angle of arc AV =1/2 (150 degrees) =75

Segments of Tangents, Secants, and Chords**
 * Section 9.5

__Objectives:__ 1. Define special cases of segments related to circles, including secant-secant secant-tangent, and chord-chord segments. 2. Develop and use theorems about measure of the segments.

__Theorem__ If two segments are tangent to a circle from the same external point, then the segments are of equal length.

__Theorem__ If two secants intersect outside a circle, the the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secants segments and its external segment. (Whole x Outside = Whole x Outside)

__Theorem__ If a secant and a tangent intersect outside a circle, then the product of the lengths of the secant segment and its external segment equals the length of the tangent segment squared.

__Example One__: Global positioning satellites are used in navigation. If the range of the satellite, AX, is 16,000 miles, what is BX?

__Solution:__ AX and BX are tangents to a circle from the same external point. By Theorem 9.5.1, they are equal.

AX= BX =16,000 miles.

__Example Two__ In the figure, EX 1.31, GX 0.45, and FX = 1.46. Find HX. Round your answer to the nearest hundredth.

__Solution:__ EX and FX are secants that intersect outside the circle. By Theorem 9.5.2, Whole x Outside = Whole x Outside.

EX x GX = FX x HX 1.31 x 0.45 = 1.46 x HX 1.46 x HX = 0.5895 HX = 0.40

Circle in the Coordinate Plane**
 * Section 9.6

__Objectives:__ 1. Develop and use the equation of a circle. 2. Adjust the equation for a circle to move the center in a coordinate plane.

__Example One:__

Given: x2 + y2 = 25 Sketch and describe the graph by finding ordered pairs that satisfy the equation. Use a graphics calculator to verify your sketch.

__Solution:__ When sketching the graph of a new type of equation, it is often helpful to locate the intercepts. To find the x-intercept, find the values of x when y = 0

x2 +0sq = 25 x2 = 25 x = (+)(-)5 Thus, the graph has two x-intercepts, (5,0) and (-5,0)

[|chords and arcs!] [|Interactive circle equations!] [|Chords] [|tangents]


 * Pictures in 9.1 and 9.2 made by me on Geometer's Sketchpad!