neje511

=Chapter 9= =

__Lessons__
9.1 Chords & Arcs 9.2 Tangents To Circles 9.3 Inscribed Angles and Arcs 9.4 Angles Formed by Secants and Tangents 9.5 Segments of Tangents, Secants & Chords 9.6 Circles in the Coordinate Plane

9.1
Circle-a set of all points in a plane that are equidistant from a given point in a plane known as the center of the circle.
 * __Vocabulary__ [|Geometry Circle Clock Link]**

Radius- a segment from the center of a circle to a point on the circle

Chord- a segment whose 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 taht divide the circle into two arcs..the points are called endpoints Semi-circle- an arc whose endpoints are endpoints of a diameter. (informally called a half-circle) Minor Arc- an arc that is shorter that a semi-circle of that circle. named by it's endpoints. Major Arc- an arc that is longer that a semicircle of that circle. named by it's endpoints and another point that lies on the arc. Central Angle- an angle in a plane of a circle whose vertex is the center of the circle. used to find measures of arcs. 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 meansure of a minor arc is the measure of it's central angle. The degree meansure of a major arc is 360° minus the degree measure of it's minor arc. the degree measure of a semicircle is 180°.

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 (2±r) Chords & Arcs Theorem- In a circle, or in congruent circles, the arcs of congruent chords are congruent.
 * __Theorems and Formulas__**

9.2
__Vocabulary__ Secant- 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 a line (tangent) in the plane of the circle that intersects

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. Converse of Tangent Theorem- If a line is perpendicular to a radius of a circle, at it's endpoint on the circle, then the line is tangent to the circle.
 * __Theorems__**

9.3
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 angle inscribed in a circle is equal to one-half the measure of the intecepted arc. Right Angle Corollary- If an inscribed angle intersepts a circle, then the angle is a right angle. Arc-Intercept Corollary- If two inscribed angles intercept the same arc, they have the same measure.
 * __Vocabulary__**
 * __Theorems & Corollaries__**

9.4
Theorem 9.4.1- If a tangent and secant, (or a chord) intersect on a circle at the point of tangency, then the measure of the angle fored is half the measure of the intersected arc. Theorem 9.4.2- The measure of an angle formed by tow secants of chords that intersect the interior of a cirlce is one-half the sum of the measures of the arcs interepted by the angle and its vertical angle. Theorem 9.4.3 - The measure of an angle formed by two secants that interset in the exterior of a circle is one-half the difference of the measures of the intercepted arcs. Theorem 9.4.4- 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 9.4.5- The measure of a tangent-tangent angles with its vertex outside the circle is one-half the difference of the measures of the intercepted arcs, or the measure of the major arc minus 180°.
 * __Theorems__**

9.5
Theorem 9.5.1- If two segments are tangent to a circle from the same external point, then the segments are of equal length. Theorem 9.5.2- If two secants intersect outside a circle, then the product of the lenths of one secant segment and its external segment equals the product of the lenths of the other secant segment and its external segment. (Whole x Outside = Whole x Outside) Theorem 9.5.3-If a secant and a tangent intersect outside a circle, then the product of the lenths of the secant segment and its external segment equals the lenth of t he tangent segment squared. (Whole x Outside = Tangent Squared) Theorem 9.5.4- If two chords intersect inside a circle, then the product of the lenths of the segments of the segments of one chord equals the product of the lengths of the segments of the other chord.
 * __Theorems__**