Section 6.3Applying Properties of Chords

Warm up for Lesson 6.3

ANSWER radius

1. DC

Tell whether the segment is best described as a radius,chord, or diameter of C.

ANSWER diameter

2. BD

Tell whether the segment is best described as a radius,chord, or diameter of C.

Warm up for Lesson 6.3

ANSWER chord

3. DE

Tell whether the segment is best described as a radius,chord, or diameter of C.

Warm up for Lesson 6.3

ANSWER chord

4. AE

Tell whether the segment is best described as a radius,chord, or diameter of C.

Warm up for Lesson 6.3

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 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.

GUIDED PRACTICE for Example 1

Use the diagram of D.

1. If mAB = 110°, find mBC

mBC = 110° ANSWER

GUIDED PRACTICE for Example 1

Use the diagram of D.

2. If mAC = 150°, find mAB

mAB = 105° ANSWER

EXAMPLE 2 Use perpendicular bisectors

SOLUTION

STEP 1 Label the bushes A, B, and C, as shown. Draw segments AB and BC .

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

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.

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

GUIDED PRACTICE for Examples 2 and 3

3. CDFind the measure of the indicated arc in the diagram.

mCD = 72°

ANSWER

GUIDED PRACTICE for Examples 2 and 3

4. DE

5. CE

Find the measure of the indicated arc in the diagram.

mCE = mDE + mCD

mCE = 72° + 72° = 144°

ANSWER

mCD = mDE.

mDE = 72°

ANSWER

EXAMPLE 4 Use Theorem 6.8

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.

In the diagram of C, QR = ST = 16. Find CU.

GUIDED PRACTICE for Example 4

6. QR

In the diagram in Example 4, suppose ST = 32, and CU = CV = 12. Find the given length.

QR = 32

ANSWER

GUIDED PRACTICE for Example 4

7. QU

In the diagram in Example 4, suppose ST = 32, and CU = CV = 12. Find the given length.

QU = 16

ANSWER

GUIDED PRACTICE for Example 4

8. The radius of C

In the diagram in Example 4, suppose ST = 32, and CU = CV = 12. Find the given length.

ANSWER The radius of C = 20

Daily Homework Quiz

For use after Lesson 10.3

1.Find the value of x in C. Explain. .

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

2.Find the value of x in C. Explain. .

ANSWER

4; In the same circle, if two chords are equidistant from the center, then they are .=~

Daily Homework Quiz

For use after Lesson 10.3

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)