Theorem 6.5

1
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.
2
Use congruent chords to find an arc measure
Example 1
Solution
chords
circles
congruent
3
Theorem 6.6
If one chord is a perpendicular bisector of
another chord, then the first chord is a diameter.
4
Theorem 6.7
If a diameter of a circle is perpendicular to a
chord, then the diameter bisects the chord and
its arc.
5
Use perpendicular bisectors
Example 2
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
C
Solution
B
Step 1
perpendicular bisectors
Step 2
Theorem 6.6
intersect
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.
Step 3
equidistant
6
Checkpoint. Complete the following exercises.

1. If mTV 121o, find mRS

By Theorem 6.5, the arcs are congruent.
mRS 121o
7
Checkpoint. Complete the following exercises.

1. Find the measures of CB, BE, and CE.

By Theorem 6.7, the diameter bisects the chord.
mCB 64o
mBE 64o
mCE 128o
8
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.
9
Use Theorem 6.8
Example 3
Solution
equidistant
GF
Use Theorem 6.8.
Substitute.
Solve for x.
6
2
So, EF 3x 3(___) ___.
10
Checkpoint. Complete the following exercises.

1. In the diagram in Example 3, suppose AB 27 and
   EF GF 7. Find CD.

By Theorem 6.8, the two chords are congruent
since they are equidistant from the center.
CD 27
11
Use chords with triangle similarity
Example 4
Theorem 6.7
SP has a given length of ___.
right angle
12
Use chords with triangle similarity
Example 4
5
SP
10
5
8
Theorem 6.8
right angle
13
Use chords with triangle similarity
Example 4

1. Find the ratios of corresponding sides.

Side-Side-Side Similarity Theorem
14
Checkpoint. Complete the following exercises.
then TP 12
NR MP 24
Since QN is the diameter and SP is
a radius, then SP 13
Side-Side-Side Similarity Theorem
15
Pg. 211, 6.3 1-26