Download Geometry of the Circle

Chapter 11
Geometry of the Circle
1
11.1 ‐ Arcs and Angles
A CIRCLE is the set ofa ll point in a plane that are equidistant
from a fixed point called the center of the circle.
RADIUS: a line segment to any point of the circle
***All radii of the same circle are congruent***
Interior of a Circle: points inside the circle
Exterior of a Circle: points outside the circle
2
Central Angles: angle whose vertex is at the center of the circle
3
Arcs‐
Intercepted Arc: a part of a circle cut off by an angle
Minor Arcs: arc is less than half of the circle
angle is less than 180
Major Arcs: arc is more than half of the circle
angle is greater than 180
Semicircle: arc that is half the circle
angle is 180 Quadrant: arc that is 1/4th of the circle
angle is 90
4
Degree Measure of an Arc‐
equal to the measure of the central angle
Minor Arc = mBC
Major Arc = mBOC
Semicircle = mAOC
5
Congruent Circles: circles with congruent radii
Congruent Arcs: arcs of the same circle or of a congruent circle that
are equal in measure
ARC ADDITION POSTULATE: If AB and BC are two arcs of the same circle having a common endpoint and no other points in common, then AB + BC = ABC and mAB + mBC = mABC
Minor Arc:
Major Arc:
Semicircle:
6
THM: In a circle or in congruent circles, congruent central angles intercept congruent arcs
(if two angles are congruent, the arcs are congruent)
(the converse) THM: In a circle or in congruent circles, congruent arcs are intercepted by congruent central angles
(if 2 arcs are congruent, the angles are congruent)
7
Let OB and OA be opposite rays of circle O, and angle BOC = 15.
Find: a) m AOC
b) mAC
c) mBC
d)mAB
e)mABC
8
Given: AB intersects CD at O, the center of the circle
Prove: AC≅BC
9
11.2 ‐ Arcs and Chords
CHORD= line segment whose endpoints are points of the circle
DIAMETER= chord that goes through the center of a circle
***a chord makes up a minor and major arc
10
In a circle or in congruent circles, ...
THM= ... congruent central angles
have congruent chords
THM= ... congruent arcs have congruent chords
THM= ...congruent chords have congruent central angles
THM= ...congruent chords have congruent arcs
11
THM: A diameter perpendicular to a chord bisects the chord and its arcs
THM: The perpendicular bisector of a chord of a circle passes through the center of the circle
12
THM: If 2 chords of a circle are congruent, they are equidistant from the center of the circle
(Converse)
THM: If 2 chords of a circle are equidistant from its center, the chords are congruent
13
Find the length of a chord 3 inches from the center of a circle whose radius measures 5 inches.
14
In a circle of radius 10, mAB=90.
a) Find the length of chord AB in radical form
b) Find the distance of AB from the center of the circle in radical form
c) Express each length from parts a and b to the nearest hundredth 15
HOMEWORK: (11.1) pg 440 #1‐7
(11.2) pg 447 #1,2,5,6,8,11,13,28
16