6.3Apply Properties of Chords Theorem 6.5 In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding

6.3 Apply Properties of Chords

Theorem 6.5In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

_____. _____ ifonly and if CDAB AB CD

CB

A D


Example 1 Use congruent chords to find an arc measureUse congruent chords to find an arc measure

In the diagram, A D, BC EF, and mEF = 125o. Find mBC.

B

C

A FD

Eo125

Solution

Because BC and EF are congruent ______ in congruent _______, the corresponding minor arcs BC and EF are __________.

chordscircles

congruent

._____EFBC So mmo125


Theorem 6.6If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.

If QS is a perpendicular bisector of TR, then ____ is a diameter of the circle.QS

T

QP

R

S


Theorem 6.7If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. F

GH

D

E

If EG is a diameter and EG DF, then HD HF and ____ ____.

GD GF


Example 2 Use perpendicular bisectorsUse perpendicular bisectors

SolutionLabel the sculptures A, B, and C. Draw segments AB and BC

perpendicular bisectorsTheorem 6.6

Journalism A journalist is writing a story about three sculptures, arranged as shown at the right. Where should the journalist place a camera so that it is the same distance from each sculpture?

A

B

C

Step 1

Draw the ____________________ of AB and BC. By _____________, these bisectors are diameters of the circle containing A, B, and C.

Step 2

Step 3 Find the point where these bisectors _________. This is the center of the circle containing A, B, and C, and so it is __________ from each point.

intersect

equidistant

6.3 Apply Properties of ChordsCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 1. If mTV = 121o, find mRS TS

R

V

66

mRS = 121o

By Theorem 6.5, the arcs are congruent.

6.3 Apply Properties of ChordsCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 2. Find the measures of

CB, BE, and CE.

By Theorem 6.7, the diameter bisects the chord.

C

B

E

D

o4x

o80 x

xx 804805 x16x

mCB = 64o

mBE = 64o

mCE = 128o


Theorem 6.8In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.

AB CD if and only if ____ ____. EF EG

F

G

E

C

B

A

D


Example 3 Use Theorem 6.8Use Theorem 6.8

Solution

Chords AB and CD are congruent, so by Theorem 6.8 they are __________ from F. Therefore, EF = _____.

equidistantGF

In the diagram of F, AB = CD = 12. Find EF.

Use Theorem 6.8.____EF GFSubstitute._______3 x 87 xSolve for x.___x 2

So, EF = 3x = 3(___) = ___.2 6

B

x3

F

G

E

C

A

D

87 x

12

12

6.3 Apply Properties of ChordsCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 3. In the diagram in Example 3,

suppose AB = 27 and EF = GF = 7. Find CD.

CD = 27

By Theorem 6.8, the two chords are congruent since they are equidistant from the center.

B

F

G

E

C

A

D

7

27

7

6.3 Apply Properties of ChordsExample 4 Use chords with triangle similarityUse chords with triangle similarity

Theorem 6.7

In S, SP = 5, MP = 8, ST = SU, QN MP, and NRQ is a right angle. Show that PTS NRQ.

M T

U

N

SR

Q

P

5

8

1. Determine the side lengths of PTS. Diameter QN is perpendicular to MP, so by ___________ QN bisects MP. Therefore,

.____2

1___

2

1PT MP 8 4 SP has a given length of ___.5

Because QN is perpendicular to MP, PTS is a __________ right angle

.______________TS and 22 SP PT 25 24 3The side lengths of PTS are SP = ____, PT = ____, and TS = ____.5 4 3

2. Determine the side lengths of NRQ. The radius SP has a length of ___, so the diameter QN = 2(___) = 2(__) = ___.


5


M T

U

N

SR

Q

P

5

8

SP 5 10By _____________ NR MP, so NR = MP = __.Theorem 6.8 8Because NRQ is a ____________, right angle

.______________RQ 22 QN NR 201 28 6The side lengths of NRQ are QN = ___, NR = ___, and RQ = ___.10 8 6

3. Find the ratios of corresponding sides.



M T

U

N

SR

Q

P

5

8

.QN

SP and ,

RQ

TS ,

NR

PT______________________________

8

4

2

16

3

2

1

10

5

2

1

Because the side lengths are proportional, PTS NRQby the ________________________________.

Side-Side-Side Similarity Theorem

6.3 Apply Properties of ChordsCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 4. In Example 4, suppose in S,

QN = 26, NR = 24, ST = SU, QN MP, and NRQ is a right angle. Show that PTS NRQ.

M T

U

N

SR

Q

P

2426

.______________RQ 22 QN NR 226 242 10

10NR = MP = 24

12

then TP = 12 Since QN is the diameter and SP isa radius, then SP = 13

13

.______________ST 22 SP TP 213 212 5

5

.QN

SP and ,

RQ

TS ,

NR

PT______________________________

24

122

110

52

1

26

13

2

1

Because the side lengths are proportional, PTS NRQby the ________________________________.

Side-Side-Side Similarity Theorem


Pg. 211, 6.3 #1-26

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6.3Apply Properties of Chords Theorem 6.5 In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding