Section 6.3
Applying Properties of Chords
Warm up for Lesson 6.3
Tell whether the segment is best described as a radius,
chord, or diameter of C.
1.
DC
ANSWER
radius
Warm up for Lesson 6.3
Tell whether the segment is best described as a radius,
chord, or diameter of C.
2. BD
ANSWER
diameter
Warm up for Lesson 6.3
Tell whether the segment is best described as a radius,
chord, or diameter of C.
3. DE
ANSWER
chord
Warm up for Lesson 6.3
Tell whether the segment is best described as a radius,
chord, or diameter of C.
4. AE
ANSWER
chord
Theorem 6.5
Theorem 6.6
Theorem 6.7
Theorem 6.8
EXAMPLE 1
Use congruent chords to find an arc measure
In the diagram, P
mFG
Q, FG
JK , and mJK = 80o. Find
SOLUTION
Because FG and JK are congruent chords in
congruent circles, the corresponding minor arcs FG
and JK are congruent.
So, mFG = mJK = 80o.
GUIDED PRACTICE
Use the diagram of
for Example 1
D.
1. If mAB = 110°, find mBC
ANSWER
mBC = 110°
GUIDED PRACTICE
Use the diagram of
for Example 1
D.
2. If mAC = 150°, find mAB
ANSWER
mAB = 105°
EXAMPLE 2
Use perpendicular bisectors
Gardening
Three bushes are arranged in a
garden as shown. Where
should you place a sprinkler so
that it is the same distance
from each bush?
SOLUTION
STEP 1
Label the bushes A, B, and C,
as shown. Draw segments AB
and BC .
EXAMPLE 2
Use perpendicular bisectors
STEP 2
Draw the perpendicular
bisectors of AB and BC By
Theorem 10.4, these are
diameters of the circle
containing A, B, and C.
STEP 3
Find the point where these
bisectors intersect. This is the
center of the circle through A,
B, and C, and so it is
equidistant from each point.
EXAMPLE 3
Use a diameter
Use the diagram of E to find the length of AC . Tell
what theorem you use.
ANSWER
Diameter BD is perpendicular to AC .
So, by Theorem 10.5, BD bisects AC ,
and CF = AF.
Therefore, AC = 2 AF = 2(7) = 14.
GUIDED PRACTICE
for Examples 2 and 3
Find the measure of the indicated arc in the diagram.
3. CD
ANSWER
mCD = 72°
GUIDED PRACTICE
for Examples 2 and 3
Find the measure of the indicated arc in the diagram.
4. DE
ANSWER
mCD = mDE.
mDE = 72°
5.
CE
ANSWER
mCE = mDE + mCD
mCE = 72° + 72° = 144°
EXAMPLE 4
Use Theorem 6.8
In the diagram of C, QR = ST = 16. Find CU.
SOLUTION
Chords QR and ST are congruent, so by Theorem 10.6
they are equidistant from C. Therefore, CU = CV.
CU = CV
2x = 5x – 9
x=3
So, CU = 2x = 2(3) = 6.
Use Theorem 6.8
Substitute.
Solve for x.
GUIDED PRACTICE
for Example 4
In the diagram in Example 4, suppose ST = 32, and
CU = CV = 12. Find the given length.
6. QR
ANSWER
QR = 32
GUIDED PRACTICE
for Example 4
In the diagram in Example 4, suppose ST = 32, and
CU = CV = 12. Find the given length.
7. QU
ANSWER
QU = 16
GUIDED PRACTICE
for Example 4
In the diagram in Example 4, suppose ST = 32, and
CU = CV = 12. Find the given length.
8. The radius of C
ANSWER
The radius of
C = 20
Daily Homework Quiz
For use after Lesson 10.3
Find the value of x in . C. Explain.
1.
ANSWER
6; If a diameter of a circle is to the chord, then the
diameter bisects the chord and its arc.
Daily Homework Quiz
For use after Lesson 10.3
Find the value of x in . C. Explain.
2.
ANSWER
4; In the same circle, if two chords are equidistant from
~.
the center, then they are =
Daily Homework Quiz
3.
For use after Lesson 10.3
Determine whether RS is a diameter.
ANSWER
Yes. Sample answer: RS is the bisector of TU by
Theorem 5.3. Then RS is a diameter of the circle by
Theorem 10.4.
Homework page 201 (1-13 odd)
page 203 (5-15 odd, 16-22 all)