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=Chapter 9 Circles= [|Chords and Arcs....]
 * __9.1 Chords and Arcs__**
 * Objectives:**
 * 1) Define a circle and its associated parts, and use them in construction.
 * 2) Define and use the degree measure of arcs.
 * 3) Define and use the length measure of arcs.
 * 4) Promce a theorem about chords and their intercepted arcs.

Circle- The set of points in a plane that are equidistant from a given point known as the center of the circle. Radius- A segment that connects the center of a circle with a point on the circles; one-haalf the diameter of a circle. Chord- A segment whose endpoints lie on a circle. Diameter- A chord the passes through the center of a circle; twice the length of the radius of the circle. Arc- An unbroken part of a circle. Endpoints- A point at an end of a segment or the starting point of a ray. Semi-Circle- The arc of a circle whose endpoints are the endpoints of a diameter. Minor Arc- An arc of a circle that is shorter than a semicircle of that circle. Major Arc- An arc of a circle that is longer than a semicircle of that circle. Central Angle- An angle formed by two rays originating from the center of a circle. Inercepted Arc- An arc whose endpoints lie on the sides of an inscribed angle. Degree Measure of Arcs- The measure of a minor arc is the measure of its central angle. The degree measure of a major arc is 360° minus the degree measure of its central angle.
 * Definitions**

Find the measures of arcs RT, TS and RTS.
 * Example One:**

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.

Find length of the arc
 * Example Two:**

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

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)


 * Arc Length:** The measure of an arc of a circle in terms of linear units, such as centimeters.


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

[|Tangent to Circles...] Objectives:**
 * __9.2 Tangents to Circles__
 * 1) Define tangents and secants of circles.
 * 2) Understand the relationship between tangents and certain radii of circles.
 * 3) Understand the geometry of a radius perpendicular to a chord of a circle.


 * Secant-** A line that intersects a circle at two points.
 * Tangent-** In a right triangle, the ratio of the length of the side opposite on acute angle to the length of the side adjacent to it.
 * Point of Tangency-** The point of intersection of a circle or sphere with a tangent line or plane.
 * Tangent Theorem-** If a line is tangent to a circle, then 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.

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

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


 * Converse of Tangent Theorem-** If a line is perpendicular to a radius of a circle at its endpoints on the circle, then the line is... to the circle.

[|Inscribed Angles and Arcs...] Objectives:**
 * __9.3 Insribed Angles and Arcs__
 * 1) Define inscribed angle and inercepted arc.
 * 2) Develop and use the inscribed Angle Throrem and its corollaries.


 * Inscribed Angle Theorem-** An inscribed Angle is an angle whose vertex lies on a circle and whose sides are chords of the circle.

Find the measure of XVY
 * Example One:**

XVY is inscribed in P and intercepts XY. By the Inscribed Angle Theroem: m XVY 1/2(45) 221/2


 * Right-Angle Corollary-** If an inscribed angle intercepts a semi circle, then the angle is a right angle.


 * Arc-Intercept Corollary-** If two inscribed angles intercept the same arc, then they have the same measure.

A person's effective field of vision is about 30 degrees. In the diagram of the amphitheater, a person sitting at point A can see the entire stage. What is the measure of B? Can the person sitting at point B view the entire stage?
 * Example Two:**

Angles A and B intercept the same arc. By Corollary 9.3.3, the angles must have the same measure, so m A= m B =30. The person sitting at point B can view the entire stage.

[|Angles Formed by Secants and Tangents...] Objectives:**
 * __9.4 Angles Formed by Secants and Tangents__
 * 1) Define angles formed by secants and tangents of circles.
 * 2) Develop and use theorems about measures of arcs intercepted by these angles.


 * Case 1-** Vertex is on circle.


 * Case 2-** Vertex is inside the circle.


 * Case 3-** Vertex is outside the circle.


 * Theorem pg. 589-** If a tangent and a secant for a chord intersect on a circle at the point of tangency, then the measure of the angle formed is... the measure of its intercepted arc.


 * Theorem 9.4.2-** 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 9.4.3-** The measure of an angle formed by two secants that intersects in the extior of a circle is... the... of the measure of the intercepted arcs.

[|Secants...] Objectives:**
 * __9.5 Segments of Tangents, Secants, and Chords__
 * 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.

are of equal length.
 * Theroem-** If two segmentsare tangent to a circle from the same external point, then the segments

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)
 * Theroem-** If two secants intersect outside a circle, the the product of the lengths of one secant segment

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

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

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

AX= BX =16,000 miles.

In the figure, EX 1.31, GX 0.45, and FX = 1.46. Find HX. Round your answer to the nerest hundredth.
 * Example Two:**

Solution: EX and FX are secants that intersect outside the circle. By Theroem 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

Objectives:**
 * __9.6 Circles in the Coordinate Plane__
 * 1) Develop and use the equation of a circle.
 * 2) Adjust the equation for a circle to move the center in a coordinate plane.

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

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 The graph has two x-intercepts, (5,0) and (-5,0)