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ARCs and chords. Geometry CP2 (Holt 12-2) K.Santos. Central Angle. Central angle ----an angle—vertex----center A B O < AOB is a central angle. Chords and Arcs. Chord —endpoints are on the circle A B

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ARCs and chords

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## ARCs and chords

Geometry CP2 (Holt 12-2) K.Santos

### Central Angle

Central angle----an angle—vertex----center

A

B

O

< AOB is a central angle

### Chords and Arcs

Chord—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 of circle)

### Arcs and their measures

Minor arc

small arc (smaller than half circle)---between 0-180

Measure = central angle

Name with two letter

Major arc---

big arc (bigger than half circle)---between 180-360

Measure = 360- minor arc

name with 3 letters

Semicircle---arc is half the circle

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—arcs of the same circle next to each other (share endpoint)

R

S O U

T

and

and

The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs.

A B

C

Add 2 little arcs together to get big arc

m = m + m

Find m and m.M

Y

40 56

DW

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:

Congruent central angles

Congruent chords

Congruent arcs

### 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 (12-2-4)

Radius (or diameter) is perpendicular to a chord----- bisects the chord and its arc.

D

A B C

F

Given: is a diameter and is perpendicular to

Then: is bisected (

is bisected (

### 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