Lesson 8-4: Arcs and Chords 1

Lesson 8-4

Arcs

and Chords

Lesson 8-4: Arcs and Chords 2

Theorem #1:

If AB CD then AB CD

127 , .Given mAB find the mCD

In a circle, if two chords are congruent then their corresponding minor arcs are congruent.

E

A B

CD

Example:

127

Since mAB mCD

mCD

Lesson 8-4: Arcs and Chords 3

Theorem #2:

sec .

If DC AB then DC bi ts AB and AB

AE BE and AC BC

120 , .If mAB find mAC

In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects the chord and its arc.

E

D

A

C

BExample: If AB = 5 cm, find AE.

52.5

2 2

120, 60

2 2

ABAE AE cm

mABmAC mAC

Lesson 8-4: Arcs and Chords 4

Theorem #3:In a circle, two chords are congruent if and only if they are equidistant from the center.

O

A B

C

DF

EExample: If AB = 5 cm, find CD.

Since AB = CD, CD = 5 cm.

CD AB iff OF OE

Lesson 8-4: Arcs and Chords 5

Try Some Sketches: Draw a circle with a chord that is 15 inches long and 8 inches from

the center of the circle. Draw a radius so that it forms a right triangle. How could you find the length of the radius?

secOD bi ts AB

8cm

15cm

O

A BD

∆ODB is a right triangle and

2 2 2

2 2 2

AB 15DB= = =7.5 cm

2 2OD=8 cmOB =OD +DBOB =8 +(7.5) =64+56.25=120.25OB= 120.25 11 cm

Solution:

x

Lesson 8-4: Arcs and Chords 6

Try Some Sketches: Draw a circle with a diameter that is 20 cm long. Draw another chord (parallel to the diameter) that is 14cm long. Find the distance from the smaller chord to the center of the circle.

14sec . 7

2 2

ABOE bi ts AB EB cm

10 cm10 cm

20cm O

A B

DC

14 cm

xE

Solution:

OB (radius) = 10 cm∆EOB is a right triangle.2 2 2

2 2 2

2

10 7

100 49 51

51

OB OE EB

X

X

X

7.1 cm