MM2G3a MM2G3a Understand and use properties of Understand and use properties of

chords, tangents, and secants as chords, tangents, and secants as an application of triangle similarity.an application of triangle similarity.

Theorem 6.5

In the same circle, or in congruent circles, two minor arcs are

congruent if and only if their corresponding chords are

congruent.

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

Theorem 6.6

If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.

If BD is a perpendicular bisector of EC, then BD is a diameter of the circle.

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 6.6, 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.

Theorem 6.7

If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

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 6.7, 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

Theorem 6.8

In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.

EXAMPLE 4

SOLUTION

Chords QR and ST are congruent, so by Theorem 6.8 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