Section 6 3 Applying Properties of Chords Warm

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 m. FG Q, FG JK , and m. JK = 80 o. Find SOLUTION Because FG and JK are congruent chords in congruent circles, the corresponding minor arcs FG and JK are congruent. So, m. FG = m. JK = 80 o.

GUIDED PRACTICE Use the diagram of for Example 1 D. 1. If m. AB = 110°, find m. BC ANSWER m. BC = 110°

GUIDED PRACTICE Use the diagram of for Example 1 D. 2. If m. AC = 150°, find m. AB ANSWER m. AB = 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 m. CD = 72°

GUIDED PRACTICE for Examples 2 and 3 Find the measure of the indicated arc in the diagram. 4. DE ANSWER m. CD = m. DE = 72° 5. CE ANSWER m. CE = m. DE + m. CD m. CE = 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 2 x = 5 x – 9 x=3 So, CU = 2 x = 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

Homework page 201 1 -13