drja227

__9.1 Chords and Arcs Objectives:__
1**. Define a [|circle] and use the degree measure of arcs.
 * __Objectives:__
 * 2.** Define a circle and its associated parts, and use them in constructions.
 * 3.** Define and use the length measure of [|arcs].
 * 4.** Prove a theorem [|chords] and their intercepted arcs.

__**Definitions:**__ __**Example One:**__ Arc mTS= 90 degrees.
 * Circle-** the set of all points in a plane that are equidistant from a given point in the plane known as the center of the circle.
 * Radius-** segment from the center
 * Chord-** segment whose endpoints line on a circle.
 * Diameter-** a chord that contains the center of a circle.
 * Arc-** an unbroken part of the circle.
 * Endpoints-** two distinct points on a circle divide the circle into two arcs.
 * Semi-circle-** an arc whose endpoints are endpoints of a diameter.
 * Minor arc-** a circle is an arc that is shorter than a semi-circle of that circle named by endpoints.
 * Major arc-** an arc that’s longer than a semi-circle of that circle named by endpoints and another point that lies on the arc.
 * Central angle-** a circle is an angle in the plane of a circle whose vertex is the center of
 * Intercepted arc-**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 360 degrees minus the degree measure of its minor arc.
 * 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...
 * L= __m__ ( 2πr )**
 * 360°**
 * Chords and Arcs Theorems-** In a circle, or in congruent circles, the arcs of congruent chords are congruent.
 * 1.**Find the measures of arcs RT, TS and RTS.
 * 2.**The measures of arc RT and arc TS are found from their central angles.
 * 3.** Arc mRT = 100 degrees.
 * 4.** 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**

__9.2 Tangents to Circles__

 * __Objectives:__**
 * 1.** Define [|tangents] and [|secants] of a circle.
 * 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.

Sectants-** A line that intersects a circle at two points.
 * __Definitions:__
 * Tangents-** A line in a plane of the circle that intersects the circle at exactly one point.
 * Point of tangency-** The point of a circle or sphere with a tangent line or plane.


 * __Theorems:__**
 * Tangent Theorem-** If a line is the tangent to 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 is bisect of 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.

Circle "D" has a radius of 6 in. and DX is 2 in. Line //DR// is perpendicular to line //AB// at point "Z". Find AB.
 * __Example:__**

__**Solution:**__ By the [|Pythagorean Theorem]: ( AZ )² + 2² = 6² ( AZ )² = 6² - 2² ( AZ )² = 32
 * AZ = 5.656854249**

__9.3 Inscribed Angles and Arcs__

 * __Objectives:__**
 * 1.** Define inscribed angle and intercepted arc
 * 2.** Develop and use the inscribed angle theorem and its corollaries.

__**Definitions:**__
 * Inscribed angle** - An angle whose vertex lies on a circle and whose sides are cords of the circle.

__**Theorems:**__
 * Inscribed angle theorem** - The measure of an angle inscirbed in a circle is equil to one 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.

A person's effective field of vision is about 40°. In the diagram of the amphitheatre, a person sitting at point "A" can see the entire stage. What is the measure of angle //B//? Can the person sitting at point "B" view the entire stage? Angles //A// and //B// intercept the same arc. By Corollary 9.3.3, the angles must have the same measure...so measure of angle //A =// the measure of angle //B...//which is equal to //=// 40°. The person sitting at point "B" can view the entire stage.
 * __Example:__**
 * __Solution:__**

__Ch 9.4 Angles Fomed by Secants and Tangents Objectives:__

 * 1.**Define angles formed by secants and tangents of circles.
 * 2.**Develop and use theorems about measures of arcs interepted by these angles.

//Case # 1-// Vertex is on the circle.

//Case # 2-// Vertex is inside the circle.

//Case # 3-// Vertex is outside the circle.


 * __Example:__**
 * Find mAVC in each figure...**

By theorem 9.3.1, mAVC= ½mAV½ (150°)...
 * A.** Angle AVC is formed by a secant and a tangent that intersect on the circle.
 * = 75°**

By theorem 9.3.2, mAVC=½ (mAC + mBD)... =½ (80° + 40°)...
 * B.** Angle AVC is formed by two secants that intersect inside the circle.
 * = 60°**

By theorem 9.3.3, mAVC=½ (mAC - mBD)... ½ (80° - 20°)...
 * C.** Angle AVC is formed by two secants that intersect outside the circle.
 * = 30°**

= =
 * Theorem 1:** If a tangent and a secant (or a chord) intersect on a circle at the point of tangency, then the measure of the angle fromed is halfh the measure of its intercepted arc.
 * Theorem 2:** The mesure of an angle formed by two secants or chords that intersect in the interior of a circle is half the sum of the mesures of the arcs intercepted by the angle and its verticle angle.
 * Theorem 3:** The measure of an angle formed by two secants that intersect in the exterior of a circle is half the difference of the measures of the intercepted arcs.
 * Theorem 4:** The measure of a secant-tangent angle with its vertex outside the circle is one-half the difference of the measures of the intercepted arcs.
 * Theorem 5:** The measure of a tangent-tangent angle with its vertex outside the circle is one-half the difference of the measures of the intercepted arcs.

__9.5 Segments of Tangents, Secants, and Chords__

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

__**Theorems:**__ If two segments are tangent to a circle from the same external point, then the segments are equal length. __**Theorem 2:**__ If two secants intersect outside a circle, then the product of the lengths of one secant segment and external segment equals the product of the lengths of the other secant segment and its external segment. __**Theorem 3:**__ 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. __**Theorem 4:**__ If two chords intersect inside a circle, then the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.
 * __Theorem 1:__**
 * ( Whole * Outside = Whole * Outside )**



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 Theorem 9.5.1, they are equal.
 * AX= BX =16,000 miles.**

__**Example Two:**__ In the figure, EX=1.45 GX =0.50, and FX = 1.51. 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 1.45 x 0.50 = 1.51 x HX 1.51 x HX = 0.725
 * HX = 0.48**

__Objectives:__

 * 1.** Develope and use the equation of a circle.
 * 2.** Adjust the equation for a circle to move the center in a coordinate plane.

**__Equations:__**

 * 1.** When the center of the circle is at the origin (0,0)
 * X² + Y² = r ²**

(h,k) is the origin.
 * 2.** When the center of the circle is not at the origin.
 * (X - h)² + (Y - k)² = r ²**

1.** A circle on the origin (0,0) and a radius of 7 x^2 + Y^2 = 7^2 (x - 6)^2 + (y - 2)^2 = 4^2
 * __Example:__
 * 2.** A circle with a center of (6,2) and a radius of 4

= =