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

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Arcs and chords

ARCs and chords

Geometry CP2 (Holt 12-2) K.Santos


Central angle

Central Angle

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

A

B

O

< AOB is a central angle


Chords and arcs

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

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 and chords

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

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

R

S O U

T

and

and


Arc addition postulate 12 2 1

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

Add 2 little arcs together to get big arc

m = m + m


Example

Example

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

Theorem 12-2-2

Within a circle or in congruent circles:

Congruent central angles

Congruent chords

Congruent arcs


Example1

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

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 (


Example2

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


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