Section 6.3 Applying Properties of Chords
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Transcript of Section 6.3 Applying Properties of Chords
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)