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