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Lesson 9-4

Arcs

and Chords

2

Theorem #1:

If AB ≅CD then ABª ≅CDª

Given mABª =127o, findthemCD.º

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

E

A B

CD

Example:

Since mABª =mCDª

mCDª =127o

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Theorem #2:

If DC ⊥ AB then DC bisectsAB and ABª .

∴ AE ≅BE and ACª ≅BCª

If mABª =120o, find mACª .

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.

AE =AB2

∴ AE =52

=2.5 cm

mACª =mABª

2,∴mACª =

1202

=60o

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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≅ ≅

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Try Some: 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

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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