Arcs and Chords

Example 1: Applying Congruent Angles, Arcs, and Chords TV  WS. Find mWS. 9n – 11 = 7n n = 22 n = 11 = 88°  chords have  arcs. Def. of  arcs Substitute the given measures. Subtract 7n and add 11 to both sides. Divide both sides by 2. Substitute 11 for n. Simplify. mTV = mWS mWS = 7(11) + 11 TV  WS

Example 2: Applying Congruent Angles, Arcs, and Chords  C   J, and mGCD  mNJM. Find NM. GD = NM  arcs have  chords. GD  NM  GCD   NJM Def. of  chords

Example 2 Continued 14t – 26 = 5t + 1 9t = 27 NM = 5(3) + 1 = 16 Substitute the given measures. Subtract 5t and add 26 to both sides. Divide both sides by 9. Simplify. t = 3 Substitute 3 for t.  C   J, and mGCD  mNJM. Find NM.

Example 3 PT bisects RPS. Find RT. 6x = 20 – 4x 10x = 20 x = 2 RT = 6(2) RT = 12 Add 4x to both sides. Divide both sides by 10. Substitute 2 for x. Simplify. RPT  SPT RT = TS mRT  mTS

Example 4  A   B, and CD  EF. Find mCD. Find each measure. 25y = (30y – 20) 20 = 5y 4 = y CD = 25(4) Subtract 25y from both sides. Add 20 to both sides. Divide both sides by 5. Substitute 4 for y. Simplify. mCD = 100 mCD = mEF  chords have  arcs. Substitute.

Find NP. Example 5: Using Radii and Chords Step 2 Use the Pythagorean Theorem. Step 3 Find NP. RN = 17 Radii of a  are . SN 2 + RS 2 = RN 2 SN = 17 2 SN 2 = 225 SN = 15 NP = 2(15) = 30 Substitute 8 for RS and 17 for RN. Subtract 8 2 from both sides. Take the square root of both sides. RM  NP, so RM bisects NP. Step 1 Draw radius RN.

Example 6 Find QR to the nearest tenth. Step 2 Use the Pythagorean Theorem. Step 3 Find QR. PQ = 20 Radii of a  are . TQ 2 + PT 2 = PQ 2 TQ = 20 2 TQ 2 = 300 TQ  17.3 QR = 2(17.3) = 34.6 Substitute 10 for PT and 20 for PQ. Subtract 10 2 from both sides. Take the square root of both sides. PS  QR, so PS bisects QR. Step 1 Draw radius PQ.

  T   U, and AC = Find PL to the nearest tenth. Example 7

P # Homework

Lesson 8-4: Arcs and Chords 13 Extra Examples or Mini Quiz: In a circle, if two chords are congruent then their corresponding minor arcs are congruent. E A B C D Example:

Lesson 8-4: Arcs and Chords 14 In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects the chord and its arc. E D A C B Example:If AB = 5 cm, find AE. Extra Examples or Mini Quiz:

Lesson 8-4: Arcs and Chords 15 In a circle, two chords are congruent if and only if they are equidistant from the center. O A B C D F E Example: If AB = 5 cm, find CD. Since AB = CD, CD = 5 cm. Extra Examples or Mini Quiz:

Lesson 8-4: Arcs and Chords 16 Draw a circle with a chord that is 15 inches long and 8 inches from the center of the circle. Draw a radius so that it forms a right triangle. How could you find the length of the radius? 8cm 15cm O A B D ∆ODB is a right triangle andSolution: x Extra Examples or Mini Quiz:

Lesson 8-4: Arcs and Chords 17 Draw a circle with a diameter that is 20 cm long. Draw another chord (parallel to the diameter) that is 14cm long. Find the distance from the smaller chord to the center of the circle. 10 cm 20cm O A B DC 14 cm x E Solution: OB (radius) = 10 cm∆EOB is a right triangle. 7.1 cm Extra Examples or Mini Quiz: