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Chapter 9

9.1

Definitions:

Circle: A circle is the set of all points if a plane that are equidistant from a given point in the plane known as the center of the circle.

Radius: A segment from the center of a circle to a point on the circle.

Chord: A segment whose endpoints lie on a circle.

Diameter: A chord that containsthe center of a circle.

Arc: An unbroken part of a circle. Any two distinct points on a circle divide the circle into two arcs.

Endpoints: The points that divide a circle into two arcs and and end a segment or a ray.

Semi-circle: An arc whose endpoints are also the endpoints of the diameter of the circle.

Minor Arc: An arc that is shorter than a semi-circle.

Major Arc: An arc that is longer than a semi-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 central angle and whose other points lie in the interior of the central 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 the circles minor arc. ( A semi-circle always equals 180 degrees)

Example 1:

Find the degree measures of arc RT, arc TS, and arc RTS if the center of the circle is point "A" and the measure of angle RAT is 100 degrees and the measure of angle TAS is 90 degrees.

The measures of arc RT and arc TS are found from their central angles so: the measure of arc RT= 100 degrees and the measure of arc TS= 90 degrees

To find the measure of arc RTS all you have to do is add the measures of arc RT and arc TS: so 100 degrees + 90 degrees = 190 degrees

And your done. (Pretty easy)

9.2

Definitions:

Secant: A line that intersects a circles at two points.

Tangent: A line that intersects a circle at exactly one point.

Point of Tangency: The point at which a tangent intersects a circle.

9.3

Definitions:

Right-Angle Corollary: If an inscribed angle intersepts 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.

9.4

Case 1: Vertex is on the circle. Theorem: 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 the measure of its intercepted arc.
 * Secant and a Tangent
 * Two Tangents

Equation: (x/2)°

= = 9.5


 * Define special cases of segments related to circles, including secant-secant, secant-tangent, and chord-chord segements.
 * Develope and use theorems about measures of the segments.

Theorem Segments formed by Tangents:
 * If two segments tangents to a circle from the sane external point, then the segments are equal.

Segements formed by Secants:

If two secants out a circle, the product of the lenght of one secant segment and its external segment equals the product of the length of one secant segment and its external equals the product. (Whole x Outside = Whole x Outside)

If a secant and a tangent intersect a circle, then the product of the lengths of the secant segment and its external segment equals the product of the length of the secant segment and its external segment equals the product. (Whole x Outside = Tangent Squared)

Segments formed by Interesting Chords:

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 one chord.

9.6

Ojectives: Develope and use the equation of a circle.

Adjust the equation for a circle to move the center in a coordinate plane.

Example: 1. Equation for a circle with a center at the origin is, x² + y² = r². r is the radius find the x and y intercepts

The intercepts are: (9,0), (-9,0) , (0,9) , (0,-9)

The equation is: x² + y² = 3²