Thursday, April 17, 2008

Geometry Notes – chapter 10
April 9, 2008

Glossary Terms
-Arc
-Arc Length
-Arc Measure
-Center of a circle
-Central angle of a circle
-Chord
-Diameter
-Intercepted Arc
-Major Arc
-Minor Arc
-Radius
-Semicircle

Theorems, Postulates, and Definitions
-Circle: is the set of all points in a plane that are equidistant from the given point in the plane known as the center of a circle.

-Radius: a segment that goes from the center to the circle.
-All radii in the same or congruent circles are congruent
-Chord: segment within a circle that intersects the circle in 2 points.
-Congruent chords are equidistant from the center
-Diameter: the longest chord of a circle and contains the center of the circle.






Radius Chord Diameter

Arcs

-A chord divides a circle into 2 arcs.

Minor Arc Major Arc





-A third point must be added to the circle in order to identify a major arc

-Central Angle: the central angle of a circle is an angle in the plane of a circle whose vertex is the center of the circle, or more simply, the central angle is the angle between 2 radii.
B
A


Arc Measurement
-Arcs are measured in 2 ways:
The degree measure of the arc: this is the measure of an angle that the arc would intercept
The length of the arc: the actual part of the circumference that makes up the arc

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 central angle.
-The degree measure of a semicircle is 180°

Theorems, Postulates, &Definitions
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:
L= M/360 (2πr)

April 14th, 2008

Inscribed angles – and inscribed angle is an angle made up of 2 chords. Since it is made up of chords, the vertex of the angle will be ON the circle.
There are 3 types of inscribed angles, Center Interior, Center Exterior, Center included.
Intercepted Arc – the minor arc defined by the 2 endpoints of chords forming an inscribed angle that are not part of the vertex of the inscribed angle.

Theorems, Postulates, and Definitions
-Inscribed Angle Theorem 10 -5: the measure of an angle inscribed in a circle is equal to half the measure of the intercepted arc (or, the measure of an intercepted arc is twice the measure of the inscribed angle.)
-Theorem 10-6: Arc-Intercepted Theorem: if 2 inscribed angles intercept congruent arcs or the same arc, the angles are congruent.
-Theorem 10-7: Right-Angle Theorem: if an inscribed angle intercepts a semicircle, then the angle is a right angle.

Inscribed Quadrilateral Theorem
-Theorem 10-8: If a quadrilateral is inscribed in a circle then it’s opposite angles are supplementary.

April 16th, 2008

Secants, Tangents, and Angle measures
-Secant-a line that intersects a circle in 2 points





-segments AB and CB are secants.
Interior Angle Theorem
-Angles that are formed by 2 intersecting chords.
D
A B
C
-Interior Angle Theorem: the measure of the angle formed by the 2 intersecting chords is equal to ½ the sum of the measures of the intercepted arcs.
Exterior Angles
-an angle formed by 2 secants, 2 tangents, or a secant and a tangent drawn from a point outside the circle.
-Exterior Angle Theorem-the measure of the angle formed is equal to ½ the difference of the intercepted arcs.

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