Arcs and Chords Bell Ringer Draw a circle

Arcs and Chords Bell Ringer Draw a circle and construct the following: center, radius, diameter, chord, secant, tangent. Holt Mc. Dougal Geometry

Arcs and Chords Objectives Apply properties of arcs. Apply properties of chords. Holt Mc. Dougal Geometry

Arcs and Chords Vocabulary central angle arc minor arc major arc Holt Mc. Dougal Geometry semicircle adjacent arcs congruent arcs

Arcs and Chords 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. Holt Mc. Dougal Geometry

Arcs and Chords Holt Mc. Dougal Geometry

Arcs and Chords Writing Math Minor arcs may be named by two points. Major arcs and semicircles must be named by three points. Holt Mc. Dougal Geometry

Arcs and Chords Adjacent arcs are arcs of the same circle that intersect at exactly one point. RS and ST are adjacent arcs. Holt Mc. Dougal Geometry

Arcs and Chords Example 2: Using the Arc Addition Postulate Find m. BD. m. BC = 97. 4 Vert. s Thm. m CFD = 180 – (97. 4 + 52 ) = 30. 6 ∆ Sum Thm. m. CD = 30. 6 m. BD = m. BC + m. CD = 97. 4 + 30. 6 = 128 Holt Mc. Dougal Geometry m CFD = 30. 6 Arc Add. Post. Substitute. Simplify.

Arcs and Chords Check It Out! Example 2 a Find each measure. m. JKL m KPL = 180° – (40 + 25)° m. KL = 115° m. JKL = m. JK + m. KL = 25° + 115° = 140° Holt Mc. Dougal Geometry Arc Add. Post. Substitute. Simplify.

Arcs and Chords Check It Out! Example 2 b Find each measure. m. LJN = 360° – (40 + 25)° = 295° Holt Mc. Dougal Geometry

Arcs and Chords Within a circle or congruent circles, congruent arcs are two arcs that have the same measure. In the figure ST UV. Holt Mc. Dougal Geometry

Arcs and Chords Holt Mc. Dougal Geometry

Arcs and Chords Example 3 A: Applying Congruent Angles, Arcs, and Chords TV WS. Find m. WS. TV WS m. TV = m. WS 9 n – 11 = 7 n + 11 2 n = 22 n = 11 chords have arcs. Def. of arcs Substitute the given measures. Subtract 7 n and add 11 to both sides. Divide both sides by 2. m. WS = 7(11) + 11 Substitute 11 for n. Simplify. = 88° Holt Mc. Dougal Geometry

Arcs and Chords Example 3 B: Applying Congruent Angles, Arcs, and Chords C J, and m GCD m NJM. Find NM. GD NM GCD NJM GD NM arcs have chords. GD = NM Def. of chords Holt Mc. Dougal Geometry

Arcs and Chords Example 3 B Continued C J, and m GCD m NJM. Find NM. 14 t – 26 = 5 t + 1 9 t = 27 t=3 NM = 5(3) + 1 = 16 Holt Mc. Dougal Geometry Substitute the given measures. Subtract 5 t and add 26 to both sides. Divide both sides by 9. Substitute 3 for t. Simplify.

Arcs and Chords Check It Out! Example 3 a PT bisects RPS. Find RT. RPT SPT m. RT m. TS RT = TS 6 x = 20 – 4 x 10 x = 20 x=2 RT = 6(2) RT = 12 Holt Mc. Dougal Geometry Add 4 x to both sides. Divide both sides by 10. Substitute 2 for x. Simplify.

Arcs and Chords Check It Out! Example 3 b Find each measure. A B, and CD EF. Find m. CD. chords have arcs. m. CD = m. EF 25 y = (30 y – 20) Substitute. Subtract 25 y from both sides. Add 20 20 = 5 y to both sides. Divide both sides by 5. 4=y Substitute 4 for y. CD = 25(4) m. CD = 100 Holt Mc. Dougal Geometry Simplify.

Arcs and Chords Holt Mc. Dougal Geometry

Arcs and Chords Example 4: Using Radii and Chords Find NP. Step 1 Draw radius RN. RN = 17 Radii of a are . Step 2 Use the Pythagorean Theorem. SN 2 + RS 2 = RN 2 SN 2 + 82 = 172 SN 2 = 225 SN = 15 Substitute 8 for RS and 17 for RN. Subtract 82 from both sides. Take the square root of both sides. Step 3 Find NP. NP = 2(15) = 30 Holt Mc. Dougal Geometry RM NP , so RM bisects NP.

Arcs and Chords Check It Out! Example 4 Find QR to the nearest tenth. Step 1 Draw radius PQ. PQ = 20 Radii of a are . Step 2 Use the Pythagorean Theorem. TQ 2 + PT 2 = PQ 2 Substitute 10 for PT and 20 for PQ. TQ 2 + 102 = 202 Subtract 102 from both sides. TQ 2 = 300 Take the square root of both sides. TQ 17. 3 Step 3 Find QR. QR = 2(17. 3) = 34. 6 Holt Mc. Dougal Geometry PS QR , so PS bisects QR.

Arcs and Chords Lesson Quiz: Part I 1. The circle graph shows the types of cuisine available in a city. Find m. TRQ. 158. 4 Holt Mc. Dougal Geometry

Arcs and Chords Lesson Quiz: Part II Find each measure. 2. NGH 3. HL 139 21 Holt Mc. Dougal Geometry

Arcs and Chords Lesson Quiz: Part III 4. T U, and AC = 47. 2. Find PL to the nearest tenth. 12. 9 Holt Mc. Dougal Geometry