dimi309

9.1

 * Circle:** A circle is a set of all points in a plane that are equidistant from a given point in the plane known as the center of the circle.[|Circles!]
 * Radius:** A segment from the center of the circle to a point on the circle.
 * Chord**: A segment whose endpoints lie on a circle
 * Diameter**: A chord that contains the center of the circle
 * Arc**: Unbroken part of a circle


 * Endpoints**: Any two distinct points on a circle divide the circle into two arcs


 * Semi-circle**: An arc whose endpoints are endpoints of a diameter. A semi-circle is informally called a half-circle. A semi-circle is named by its endpoints and another point that lies on the arc


 * Minor arc**: A minor arc lof a circle is an arc that is shorter than a semicircle of that circle. A minor arc is named by its endpoints.


 * Major arc**: 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 mother point that lies on the arc.
 * Central Angle**: A central angle of a circle is 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 and whose points lie in the interior of the angle is the intercepted arc of the central angle.

Examples: Find the measures of all the arcs on this circle. Arc AB is 144.33 because it is also the central angle Arc BC is 84.42 because it is also the central angle Arc AC is 131.25 because a circle has 360 degrees and to find Arc AC you just take 360 - Arc AB+Arc BC
 * 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 minus the degree measure of its minor arc. The degree measure of a semicircle is 180.

9.2
[|Secant]
 * Secant:** To a circle is a line that intersects the circle at two points.
 * Tangent**: A line in the plane of a circle that intersects the circle at exactly one point
 * Point of Tangency**: The tangent line intersects the circle at this point.


 * Tangent Theorem**: If a line is tangent to a cricle, then the line is perpendicular to a radius of a 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 its endpoint on the circle, then the line is tangent to the circle.

Examples: radius is 6, chord CB is 8, can you find the length of the chord's segments when it is perpendicular to the radius? Radius and chord theorem states that a radius that is perpendicular to a chord in a circle, then the radius bisects it So segment CE is equal to segment BE and they are both 4

Inscribed Angle:** Is an angle whose vertex lies on a circle and whose sides are chords of the circle [|inscribed angle]
 * 9.3
 * Inscribed Angle Theorem**: The measure of an angle inscribed in a circle is equal to half the measure of the intercepted arc.
 * Right Angle corollary:** If an inscribed angle intercepts a semi-circle, then the angle is a right angle.

Examples: length of arc CA is 50, find measure of angle ABC It is 25 because it is an inscribed angle and it is always half of its corresponding arc.
 * Arc-Intercept corollary:** If two inscribed angles intercept the same arc, then they have the same measure

Vertex on Circle: Secant and Tangent 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 half the measure of its intercepted arc.
 * 9.4

[|Two secants theorems]
 * Vertex Inside Circle: Two Secants Theorem:** The measure of an angle formed by two secants or chords that intersect in the interior of a circle is half the sum of the measures of the arcs intercepted by the angle and its vertical.

Examples: Arc DE is 50 Arc AB is 60 What is the measure of angle ACB: Angle ACB= (Arc DE + Arc AB)/ 2 Angle ACB= 55
 * Vertex Outside Circle: Two Secants Theorem:** The measure of an angle formed by two secants that intersect in the exterior of a circle is half the sum of the measures of the intercepted.

Tangent Segment:** Touches one side of the circle.
 * 9.5


 * Secant Segment:** Touches the sides of the circle.


 * External Secant Segment:** Outside part of secant.


 * Chord:** Segment in circle, not going through the center.


 * Segments Formed by Tangents Theorem:** If two segments are tangent to a circle from the same external point, then the segments are equal.


 * Segments Formed by Secants Theorem:** If two secants intersect outside a circle, the product of the lengths of one secant segment and its external segment equals the other.


 * Segments Formed by a Secant and a Tangent Theorem:** 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 tangent squared.

Examples: Segment AD= 6 Segment DE=2 Segment CE=3 Find the length of the other segments Part 1 * Part 2=Part 1 * Part 2 2*3=6 2*AE=BE*3 AE=3 BE=2 Equation for a Circle:** X^2 + Y^2 = r^2
 * Segments Formed by Intersecting Chords Theorem:** 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.
 * 9.6

[|equation for circle info]
 * Equation for a Circle not at the Origin:** (X-h)^2 + (Y-k)^2 = r^2

Examples: A circle has a radius of 7. Find the X and Y intercepts of the circle X^2+Y^2=r^2 0+Y^2=7^2 Y^2=49 Y intercept= +/-7

X^2+0=49 X^2=49 X intercept = +/-7