Geometry Section 10.3

Coins You notice that the word LIBERTY on the heads side of a quarter makes about the same arc as the words QUARTER DOLLAR on the tails side. Explain how you can use a straight ruler to check whether this is true.

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.

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 .

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. SOLUTION Diameter BD is perpendicular to AC . So, by Theorem 10.5, BD bisects AC , and CF = AF. Therefore, AC = 2 AF = 2(7) = 14.

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.