arcs and chords geometry cp1 (holt 12-2) k.santos
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ARCS AND CHORDSGeometry CP1 (Holt 12-2) K.Santos
Central Angle
Central angle----an angle whose vertex is the center of a circle.
A
B
O
< AOB is a central angle
Chords and Arcs
Chord—is a segment whose endpoints are on the circle
A
B
Arc—is an unbroken part of a circle consisting of two points called the endpoints and all the points on the circle between them (curved part)
Arcs and their measures
Minor arc---an arc whose points are on or in the interior of a central angle
small arc---between 0-180
Measure = central angle
Name with two letters
Major arc---is an arc who points are on or in the exterior of a central angle
big arc---between 180-360
Measure = 360- minor arc
name with 3 letters
Semicircle---arc that has its endpoints on a diameter
Measure = 180
name with 3 letters
Arcs M
N
O
P
Minor Arcs: Major Arcs: Semicircle:
If m<MON = 50 find m, mand m
m = 50
m = 180-50 = 130
m= 360 – 50 = 310
Adjacent Arcs
Adjacent Arcs—arcs of the same circle that have exactly one point in common.
R
S O U
T
and
and
Arc Addition Postulate 12-2-1
The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs.
A B
C
m = m + m
Example
Find m and m. M
Y
40 56
D W
X
m = m + m
m = 40 + 56
m = 96
m= m + m
m= 56 + 180
m= 236
Theorem 12-2-2
Within a circle or in congruent circles:
(1) Congruent central angles have congruent chords
(2) Congruent chords have congruent arcs
(3) Congruent arcs have congruent central angles
Example
Find each measure. V
(9x – 11)
. Find m. W
T
(7x + 11)
S
9x – 11= 7x + 11 (congruent arcs---congruent chords)
2x -11 = 11
2x = 22
x = 11
WS = 7x + 11
WS = 77 + 11 = 88
Theorem 12-2-3
In a circle, if a radius (or diameter) is perpendicular to a chord then it bisects the chord and its arc.
D
A B C
F
Given: is a diameter and is perpendicular to
Then: is bisected (
is bisected (
Theorem 12-2-4
In a circle, the perpendicular bisector of a chord is a radius (or diameter). D
A B C
F
Given: is a perpendicular bisector
Then: is a diameter of the circle
Example
Find the value of x.
5 x
5
6
Chords are equidistant from the center so the chords are congruent
Bottom chord is 2(6) = 12, so the chord on the right is also 12. Thus, x = 12
Example
Find the length of x. 12
x
20
Diameter is bisected (2 radii): ½ (20) =10
Chord is bisected: ½ (12) = 6
Use Pythagorean Theorem:
= +
100 = 36 +
64 =
= x
x = 8