1. Tell whether the segment is best described as a radius,
chord, or diameter of C.
1. DC
radius
2. BD
diameter
3. DE
chord
4. AE
chord
Solve
5. 4x = 8x – 12.
x = 3
6. 3x + 2 = 6x – 4.
x = 2

2. Use properties of segments that intersect circles.
Target
Use properties of segments that intersect circles.
You will…
Use relationships of arcs and chords in a circle.

3. Vocabulary
Chords and Arcs
Theorem In the same circle or congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.
Theorem In the same circle or congruent circles, two chords are congruent if and only if they are equidistant from the center.

4. Vocabulary
Chords and Arcs
Theorem 10.4 – If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.
Theorem 10.5 – If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

5. EXAMPLE 1
Use congruent chords to find an arc measure
In the diagram, P Q, FG JK , and mJK = 80o.
Find mFG .
SOLUTION
Because FG and JK are congruent chords in congruent circles, the corresponding minor arcs FG and JK are congruent.
So, mFG = mJK = 80o.

6. GUIDED PRACTICE
for Example 1
Use the diagram of D.
1. If mAB = 110°, find mBC
mBC = 110°
ANSWER
2. If mAC = 150°, find mAB
mAB = 105°
ANSWER

7. EXAMPLE 2
Use perpendicular bisectors
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?
Gardening
SOLUTION
STEP 1
Label the bushes A, B, and C, as shown. Draw segments AB and BC .

8. 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.

9. EXAMPLE 3
Use a diameter
Use the diagram of E to find the length of AC Tell what theorem you use.
Diameter BD is perpendicular to AC . So, by Theorem 10.5, BD bisects AC , and CF = AF. Therefore, AC = 2 AF = 2(7) = 14.
ANSWER

10. GUIDED PRACTICE
for Examples 2 and 3
Find the measure of the indicated arc in the diagram.
mCD = 72°
ANSWERS
3. CD
4. DE
mCD = mDE.
mDE = 72°
5. CE
mCE = mDE + mCD
mCE = 72° + 72°
= 144°

11. In the diagram of C, QR = ST = 16. Find CU.
EXAMPLE 4
Use Theorem 10.6
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
Use Theorem 10.6.
2x = 5x – 9
Substitute.
x = 3
Solve for x.
So, CU = 2x = 2(3) = 6.

12. GUIDED PRACTICE
for Example 4
In the diagram in Example 4, suppose ST = 32, and CU = CV = 12. Find the given length.
QR = 32
ANSWERS
6. QR
7. QU
QU = 16 8.
The radius of C
r = 20