joma616

[|Chords and arcs link]
 * 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.


 * Definitions**
 * 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.

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

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 and circles] 1. Define tangents and secents 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.
 * 9.2 Tangents to Circles**
 * Objectives: [|]**

Secant-** A line that intersects a circle at two points.
 * Definitions:
 * 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.

If a line is tangent to a circle, then the line is perpendicular to a radius of the circle drawn to the point or tangency.
 * Tangent Theorem-**

A radius that is perpendicular to a Chord of a circle bisects the chord.
 * Radius and Chord 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.
 * Converse of the Tangent Theorem-**

P has a radius of 5 in. and PX is 3 in. PR is perpendicular to AB at point X Find AB
 * Example** **One-**

Solution: By the Pythagorean Theorem: (AX)sq+3 sq=5 sq (AX)sq= 5 sq- 3 sq (AX)sq= 16 AX= 4 By the Radius and Chord Theorem, PR bisect AB, so BX AX 4 Therefore, AB AX + BX =4+4 8

[|] 1. Define inscribed angle and intercepted arc. 2. Develope and use the Inscribed Angle Theorem and its corollaries. [|angles and arcs]
 * 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.

Find the measure of XVY
 * Example One**

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

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-**

Solution: 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.

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

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

1. Define angles formed by secants and tangents of circle. 2. Develop and use theroems about measure of arcs intercepted by these angles.
 * 9.4 Angles Formed by Secants and Tangents**
 * Objectives:**

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.
 * Theroem-**

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.
 * Theroem-**

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.
 * Theroem-**


 * 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.
 * Objectives:**

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

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)
 * Theroem**

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.
 * Theroem**

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:** 1. Develop and use the equation of a circle. 2. Adjust the equation for a circle to move the center in a coordinate plane. [|circles and coordinates] 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.
 * 9.6 Circle in the Coordinate Plane
 * Example One**

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)