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6.3
Apply 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 chords are congruent.
B C
A
D
AB  _____.
AB  CD if and only if _____
CD
6.3
Apply Properties of Chords
Example 1 Use congruent chords to find an arc measure
E
B
o
In the diagram, A  D,
125
o
BC  EF, and mEF = 125 .
D
Find mBC.
F
A
Solution
C
Because BC and EF are congruent ______
chords in
circles the corresponding minor arcs
congruent _______,
congruent
BC and EF are __________.
125 .
So mBC  mEF  _____
o
6.3
Apply Properties of Chords
Theorem 6.6
If one chord is a perpendicular bisector of
another chord, then the first chord is a diameter.
T
S
P
Q
R
If QS is a perpendicular bisector of TR,
then ____
QS is a diameter of the circle.
6.3
Apply Properties of Chords
Theorem 6.7
If a diameter of a circle is perpendicular to a
chord, then the diameter bisects the chord and
its arc.
F
E
H G
D
If EG is a diameter and EG  DF, then
HD  HF and ____
GD  ____.
GF
6.3
Apply Properties of Chords
Example 2 Use perpendicular bisectors
Journalism A journalist is writing a story about
three sculptures, arranged as shown at the right. A
Where should the journalist place a camera so
that it is the same distance from each sculpture?
C
Solution
B
Step 1 Label the sculptures A, B, and C.
Draw segments AB and BC
perpendicular bisectors of AB and
Step 2 Draw the ____________________
BC. By _____________,
Theorem 6.6 these bisectors are
diameters of the circle containing A, B, and C.
intersect
Step 3 Find the point where these bisectors _________.
This is the center of the circle containing A, B,
and C, and so it is __________
equidistant from each point.
6.3
Apply Properties of Chords
Checkpoint. Complete the following exercises.
S T
1. If mTV = 121o, find mRS
6
6
R
V
By Theorem 6.5, the arcs are congruent.
o
mRS = 121
6.3
Apply Properties of Chords
Checkpoint. Complete the following exercises.
2. Find the measures of
CB, BE, and CE.
C
4x o
B
D
80  x 
o
E
By Theorem 6.7, the diameter bisects the chord.
4 x  80  x
5x  80
x  16
mCB = 64o
mBE = 64o
mCE = 128o
6.3
Apply Properties of Chords
Theorem 6.8
In the same circle, or in congruent circles, two
chords are congruent if and only if they are
equidistant from the center.
C
A
G
E
F
B
D
AB  CD if and only if ____
EG
EF  ____.
6.3
Apply Properties of Chords
Example 3 Use Theorem 6.8
In the diagram of F,
AB = CD = 12. Find EF.
Solution
A
12
G
F
D
E
12
B
7x  8
3x
C
Chords AB and CD are congruent, so
equidistant
by Theorem 6.8 they are __________
GF
from F. Therefore, EF = _____.
Use Theorem 6.8.
EF  ____
GF
3x  _______
7x  8
Substitute.
2
x  ___
Solve for x.
6
So, EF = 3x = 3(___)
2 = ___.
6.3
Apply Properties of Chords
Checkpoint. Complete the following exercises.
27
A
3. In the diagram in Example 3,
suppose AB = 27 and EF = GF = 7.
Find CD.
G
F
D
B
E
7
7
C
By Theorem 6.8, the two chords are
congruent since they are equidistant
from the center.
CD = 27
6.3
Apply Properties of Chords
Example 4 Use chords with triangle similarity
In S, SP = 5, MP = 8, ST = SU, QN
M
MP,
and NRQis a right angle.
U
Show that PTS  NRQ.

R
1. Determine the side lengths of  PTS.
Diameter QN is perpendicular to MP,
so
Theorem 6.7
by ___________
QN bisects MP. Therefore,
8
N
T
S
P
5
Q
1
1
5
4 . SP has a given length of ___.
PT  ___
8   __
MP  __
2
2
right angle
Because QN is perpendicular to MP, PTS is a __________
and TS 
___
SP   ___
PT 
2
2
3.
 ___
5  ___
4 2  __
2
The side lengths of  PTS are
SP = ____,
5 PT = ____,
3
4 and TS = ____.
6.3
Apply Properties of Chords
Example 4 Use chords with triangle similarity
In S, SP = 5, MP = 8, ST = SU, QN
M
MP,
and NRQis a right angle.
U
Show that PTS  NRQ.

2. Determine the side lengths of  NRQ. R
5 so
The radius SP has a length of ___,
SP = 2(__)
10
the diameter QN = 2(___)
5 = ___.
8
Theorem 6.8 NR  MP, so NR = MP = __.
By _____________
right angle
Because NRQ is a ____________,
2
2
2
2
QN
RQ  ___   NR
___   1
___
6.
8  __
0  ___
The side lengths of  NRQ are
QN = 10
___, NR = ___,
8 and RQ = ___.
6
8
N
T
S
Q
P
5
6.3
Apply Properties of Chords
Example 4 Use chords with triangle similarity
In S, SP = 5, MP = 8, ST = SU, QN
MP,
and NRQis a right angle.
Show that PTS  NRQ.

8
N
T
M
P
U
S
5
R
3. Find the ratios of corresponding sides.
Q
PT 4 1 TS 3 1
SP 5 1
 
,
  , and


.
6 _____
2
_____
8 _____
2 RQ _____
2
_____
NR _____
QN 10
Because the side lengths are proportional, PTS
Side-Side-Side Similarity Theorem
by the ________________________________.
 NRQ
6.3
Apply Properties of Chords
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.
N
U
26
S
R
2
2
12
13
10
___
___  __
SP   ___
5.
TP   ___
13  12
2
2
2
2
QN
___   NR
___   ___

___
__ .
26 24  10
2
5
24
NR = MP = 24 then TP = 12
Since QN is the diameter and SP is
a radius, then SP = 13
ST 
T
M
2
Q
RQ 
PT 12 1 TS 5 1
SP 13 1
 
,
  , and


.
2
_____
_____
_____
_____
2 RQ 10
2
_____
_____
NR 24
QN 26
Because the side lengths are proportional, PTS
Side-Side-Side Similarity Theorem
by the ________________________________.
 NRQ
P
6.3
Apply Properties of Chords
Pg. 211, 6.3 #1-26