Warm-UpFind the measure of each arc of circle Z.

a. mBD

b. mACE

c. mDEB

d. mABC

140°

230°

130°Z

D

55°

BA

105°90°

75°

35°

C

E

270°

Section 10 – 3 & Section 10 – 6

Apply Properties of Chords

Theorem 10-3In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

A

B C

D

Theorem 10-4If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.

S P

T

Q

R

Theorem 10-5If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

S P

T

Q

R

Theorem 10-6In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.

A

P

C

G

DB

F

Theorem 10-14 Segments of Chords Theorem

If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

D

A

C

B

E

EA • EB = EC • ED

Example 1In circle R, AB = CD and mAB = 108°. Find mCD.

Since AB and CD are = chords, then AB = CD.

mCD = 108°

R

B

108°

CA

D

~

~~

Example 2Use the diagram of Circle C to find the length of BF. Tell what theorem you used.

10 units

Theorem 10-5: If a diameter is perpendicular to a chord than the diameter bisects the chord and its arc.

C

G

A

BDF

10

Example 3In the diagram of Circle P, PV = PW, QR = 2x + 6 and ST = 3x – 1. Find QR.

Use Theorem 10-6

x = 7

QR = STR

V S

PQ

T

W 2x + 6 = 3x – 1

QR: 2(7) + 6 =

QR = 20 units

Example 4Find RT and SU.

Use Theorem 10-14 SV • VU = RV • VT

x • 4x = (x + 1) • 3xx

U

SR

Tx + 1 V 3x

4x 4x2 = 3x2 + 3xx2 = 3xx = 3

RT = (x + 1) + 3x RT = (3 + 1) + 3(3) RT = 13

SU = x + 4x SU = 3 + 4(3) SU = 15

HomeworkSection 10-3

Page 667 – 668 (3 – 11, 15, 18 – 20)

Section 10-6Page 692 – 693 (3, 4, 13, 16)