Wednesday, August 1, 2007

Geometry Chapter 10.2 Arcs and Chords,

Chapter 10.2 Arcs and Chords:

I)Vocabulary:
A) Central angle - in a plane, an angle whose vertex is the center of a circle.
B) Minor Arc - if the measure of a central angle APB is less than 180º, then A and B and the points of circle P in the interior of angle APB form a minor arc of the circle.
C) Major Arc - The points A and B and the points of circle P in the exterior of angle APB for a major arc. OR if the measure of a central angle APB is greater than 180º and less then 360º, then A and B and the points of circle P in the interior of ∠APB form a major arc of the circle.
D) Semicircle - if the endpoints of an arc are the endpoints of a diameter.
E) To name a minor arc, use the endpoints of the arc.
Example: given a minor arc on circle C with endpoints A and B, its name is arc AB and notation is the letters AB with a small arc over the two letters.
F) To name a major arc, use the endpoints and a point on the circle between the endpoints.
Example: given a major arc on circle C with endpoints A and B and a point D between A and B, its name is arc ADB.
G) Measure of a minor arc or major arc - is defined to be the measure of its central angle .
Example: measure of minor arc AB = measure of angleACB
II) Theorems:
1) Arc Addition Postulate or Arc Partition Postlate - the measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.
Example: Arc ABC = arc AB + arc BC
2) Congruent arcs - two arcs of the same circle or of congruent circles that have the same measure.
3) In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.
4) If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.
5) If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.
6) In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.