Circles Unit 6: Lesson 2 Arcs and Chords Holt Geometry Texas ©2007

Objectives and Student Expectations TEKS: G.1A, G2B, G8C, G9C The student will develop and awareness of the structure of a mathematical system. The student will make conjectures about lines, angles, and circles, choosing a variety of approaches such as coordinate, transformational, and axiomatic. The student will be expected to use Pythagorean Theorem. The student will test conjectures about the properties and attributes of circles and the lines that intersect.

A central angle is an angle whose vertex is the center of a circle. An arc is an unbroken part of a circle consisting of two points called the endpoints and all the points on the circle between them.

Minor arcs may be named by two points. Writing Math Minor arcs may be named by two points. Major arcs and semicircles must be named by three points.

Example: 1 The circle graph shows the types of grass planted in the yards of one neighborhood. Find mKLF. mKLF = 0.65 (360°) 234 mKLF =

Adjacent arcs are arcs in the same circle that intersect at exactly one point. RS and ST are adjacent arcs.

Congruent arcs are two arcs with the ______________ angle measure. same In the figure ST  UV.

Example: 2 Find mBD in circle F. mBC = 97.4 Vert. s Thm. mCFD = 180 – (97.4 + 52) = 30.6 ∆ Sum Thm. mCD = 30.6 mCFD = 30.6 mBD = mBC + mCD Arc Add. Post. = 97.4 + 30.6 Substitute. Simplify. = 128

Example: 3 TV  WS. Find mWS and mTV. TV  WS  chords have  arcs. mTV = mWS Def. of  arcs 9n – 11 = 7n + 11 Substitute the given measures. 2n = 22 Subtract 7n and add 11 to both sides. n = 11 Divide both sides by 2. mWS = 7(11) + 11 Substitute 11 for n. mWS = 88° Simplify. mTV = 9(11) - 11 Substitute 11 for n. mTV = 88° Simplify.

Example: 4 C  J, and mGCD  mNJM. Find NM and DG. GCD  NJM GD  NM  arcs have  chords. GD  NM GD = NM Def. of  chords

Example: 4 continued C  J, and mGCD  mNJM. Find NM. 14t – 26 = 5t + 1 Substitute the given measures. 9t = 27 Subtract 5t and add 26 to both sides. t = 3 Divide both sides by 9. NM = 5(3) + 1 Substitute 3 for t. NM = 16 Simplify. DG = 14(3) - 26 Substitute 3 for t. DG = 16 Simplify.

Example: 5 PT bisects RPS. Find RT and TS. RPT  SPT mRT  mTS RT = TS 6x = 20 – 4x Substitute given for RT and TS 10x = 20 Add 4x to both sides. x = 2 Divide both sides by 10. RT = 6(2) = 12 Substitute 2 for x and simplify. TS = 20 - 4(2) = 12 Substitute 2 for x and simplify.

Example: 6 Find each measure. A  B, and CD  EF. Find mCD and mEF. mCD = mEF  chords have  arcs. Substitute. 25y = (30y – 20) Subtract 25y from both sides. Add 20 to both sides. 20 = 5y 4 = y Divide both sides by 5. CD = 25(4) Substitute 4 for y. mCD = 100 Simplify. EF = 30(4) - 20 Substitute 4 for y. mEF = 100 Simplify.

Example: 7 Find NP. Step 1 Draw radius RN. RN = 17 Radii of a  are . Step 2 Use the Pythagorean Theorem. SN2 + RS2 = RN2 SN2 + 82 = 172 Substitute 8 for RS and 17 for RN. SN2 = 225 Subtract 82 from both sides. SN = 15 Take the square root of both sides. Step 3 Find NP. NP = 2(15) = 30 RM  NP , so RM bisects NP.

Example: 8 Find QR to the nearest tenth. Step 1 Draw radius PQ. Radii of a  are . Step 2 Use the Pythagorean Theorem. TQ2 + PT2 = PQ2 TQ2 + 102 = 202 Substitute 10 for PT and 20 for PQ. TQ2 = 300 Subtract 102 from both sides. TQ  17.3 Take the square root of both sides. Step 3 Find QR. QR = 2(17.3) = 34.6 PS  QR , so PS bisects QR.

Extra Examples

Find each measure. 1. NGH 2. HL 3. T  U, and AC = 47.2. Find PL to the nearest tenth.