ch+9

=__**9.1 - Chords and Arcs**__=


 * __Objectives__**
 * 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.


 * Circle -** All points in a plane that are the same distance from a point (the center of the circle)
 * Radius** - A segment from the center of the circle to a point on a circle
 * Chord** - A segment whose endpoints line on a circle.
 * Diameter** - A chord that contains the center of a 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 of the angle.

__**Blue boxes**__ : An __arc__ is an unbroken part of the circle. Any two distinct points [called enpoints] on a circle divide the circle into two arcs. A __semicircle__ is an arc whose endpoints are endpoints of a diameter. The measure of a semicircle is 180 degrees. A __minor arc__ of a circle is an arc that is shorter than a semicircle of that circle. A minor arc is named by its endpoints. The degree measure of a minor arc is the measure of its central angle. A __major arc__ of a circle is an arc that is longer than a semicircle of that circle. A major arc is named by its endpoints and another point that lies on the arc. The degree measure of a major arc is 360 minus the degree measure of its minor arc.

In a circle, or in congruent circles, the arcs of congruent chords are __congruent__. In a circle or in congruent circles, the chords of congruent arcs are __congruent__.
 * Chords and Arcs Theorem**
 * The Converse of the Chords and Arcs Theorem**

Arc Length //R// is the radius of a circle, //M// is the degree measure of an arc of the circle and //L// is the length, the arc is given by: L = M / 360° (2πR)



Find the arc length when the radius = 100 Remember that you need the circumference of the circle by using the formula C = 2πR So we plug in 100 to the forumla: Length of the arc = m/360 (2π * 100) Length of the arc= m/360 (200π)
 * Section Example:**

=__9.2 - Tangents to Circles__=


 * __Objectives__**
 * 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**- Line that intersects the circle at two points
 * Tangent-** Line in theplane of the circle that intersects the circle at exactly one point
 * Point of tangency-** The point a tangent intersects a circle

Tangent theorem: If a line is tangent to a circle then te 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 endpoint on circle then the line is tangent to the circle Theorem: The perpendicular bisector of a cord just passes through the center of the circle.
 * __Blue Boxes__** :



The second rope from the right looks like it's going through the ferris wheel, making a **secant** line. The rope on the way right looks like it's outside, but still touching, the ferris wheel making a **tangent** line. And the ropes on the left are outside the ferris wheel, with **zero points of intersection**.

AX² + 5² = 6² AX² = 6² - 5² AX² = 36 - 25 AX² = 9 AX = 3
 * Section Example:**

=__9.3 - Inscribed Angles and Arcs__=


 * __Objectives__**
 * Define incribed 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 inscribed is equal to half the measure of the intercepted arc. Right Angle Corollary: If an inscribed angle intercepts a simicircle, then the angle is a right angle. Arc-Intercepted Corollary: If two inscribed angles intercept the same arc, then they have the same measure.
 * __Blue Boxes__**



Picture made on Paint program. < ABC is an inscribed angle because the vertex lies on the circle and the sides are chords of the circle.

The measure of <ABC = 60° Arc XY intercepts <ABC. Find the measure of arc XY.
 * Section Example:**

m<ABC = 1/2 arc XY = 1/2 (60) Arc XY = 30°
 * Answer:**

=__**9.4 - Angles Formed By Secants and Tangents**__=


 * __Objectives__**
 * Define angles formed by secants and tangents of circles
 * Develop and use theorems about measures of arcs intercepted by these angles.

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. The measure of an angle formed by two secants or chords that intersect in the interior of a circle is __one-half__ the __sum__ of the measures of the arcs intercepted by the angle and its vertical angle. The measure of an angle formed by two secants that intersect in the exterior of a circle is __one-half__ the __difference__ of the measures of the intercepted arcs The measure of a secant-tangent angle with its vertex outside the circle is __one-half the difference__. The measure of a tangent-tangent angle with its vertex outside the cirlce is __one-half the difference of the__ __measures of the intercepted arcs, or the measure of the major arc minus 180__°
 * __Theorems__**




 * Section Example:**

The measure of arc AB = 60° The measure of arc DC = 110° Find the measure of the space between A & B.


 * Answer:** 85°

=**__9.5 - Segments of Tangents, Secants, and Chords__**=


 * __Objectives__**
 * 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.

If two segments are tangent to a circle from the same external point, then the segments __are of equal length__. If two secants intersect outside a circle, the product of the lenghts of one secant and its external segment equals the __product of the lengths of the other secant segment and its external segment__. ( Whole X Outside = Whole X Outside ) 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. ( Whole X Outside = Tangent Squared ) If two chords intersect inside a circle, then the product of the lengths of the segments of one chord equals the product of the lenghts of the segments of the other chord.
 * Theorems**

Made on Paint. Segment AD is tangent. Segment AC is secant. Segment BC is a chord.

AB and CD are secant lines that intersect outside the circle at EF. Find EF when: AB = 6 CD = 5 So we'll use: Whole X Outside = Whole X Outside
 * Section Example:**

=__9.6 - Circles in the Coordinate Plane__=


 * __Objectives__**
 * Develop and use the equation of a circle.
 * Adjust the equation for a circle to move the center in a coordinate plane.



x² + y² = 100 x² + 0² = 100 x² = ±10
 * Section Example:**

So, to graph this, we start at the point 0,0. From there, graph the following points: (10, 0) (-!0, 0) (0, 10) and (0, -10)

Extra Help Sites: [|Chords] [|Inscribed & Central Angle Definitions / Examples] [|Inscribed Angles Theorem] [|Major & Minor Arcs] [|Secants]