Lesson 6.2 Properties of Chords

Homework: Lesson 6.2/1-12

Objective: Discover properties of chords of a circle

What is a chord? A chord is a segment with endpoints on a circle.

Any chord divides the circle into two arcs. A diameter divides a circle into two semicircles. Any other chord divides a circle into a minor arc and a major arc.

Chord Arcs Conjecture

In the same circle, two minor arcs are congruent if and only if their corresponding chords are congruent.

CB

A

IFF

IFF

and

G

and

Perpendicular Bisector of a Chord Conjecture

If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

E

D

G

FH

JK is a diameter of the circle.

J

L

K

M

Perpendicular Bisector to a Chord ConjectureIf one chord is a perpendicular bisector of another chord, then the first chord passes through the center of the circle and is a diameter.

JK is a diameter of the circle.

J

L

K

M

If one chord is a perpendicular bisector of another chord, then the

first chord is a diameter.

E

D

G

FDG

GF

, DE EF

Ex. 4: Using Chord Arcs Conjecture

2x = x + 40 x = 40

BA

C

2x°(x + 40)°D

A

C

B

Ex. 5: Finding the Center of a Circle

• Perpendicular bisector to a chord can be used to locate a circle’s center as shown in the next few slides.

Step 1: Draw any two chords that are not parallel to each other.

Ex. 5: Finding the Center of a Circle

Step 2: Draw the perpendicular bisector of each chord. These are the diameters.

Ex. 5: Finding the Center of a Circle

Step 3: The perpendicular bisectors intersect at the circle’s center.

center

Chord Distance to the Center Conjecture

F

G

E

B

A

C

D

F

G

E

B

A

C

D

AB CD if and only if EF EG.

Ex. 7: AB = 8; DE = 8, and CD = 5. Find CF.

58

8 F

G

C

E

D

A

B

2x = x + 40 x = 40

BA

C

2x°

(x + 40)°D

Ex.4: Solve for the missing sides.

A

B

D C

7m

3m

BC =

AB =

AD ≈

7m

14m

7.6m

.AC

Ex.6: QR = ST = 16. Find CU.

x = 3

Ex 7: AB = 8; DE = 8, and CD = 5. Find CF.

58

8 F

G

C

E

D

A

B

CG = CF

CG = 3 = CF

Ex.8: Find the length of Tell what theorem you used. .BF

BF = 10

Diameter is the perpendicular bisector of

the chordTherefore, DF = BF

Ex.9: PV = PW, QR = 2x + 6, and ST = 3x – 1. Find QR.

Congruent chords are equidistant from the center.

Congruent chords intercept congruent arcs

Ex.11:

Congruent chords are equidistant from the center.