10.2. Find Arc Measures. For Your Notebook. } RT is a diameter, so C RST is a semicircle, and m C RST Find measures of arcs KEY CONCEPT

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1 10.2 Find rc Measures efore ou found angle measures. Now ou will use angle measures to find arc measures. Why? o you can describe the arc made by a bridge, as in Ex. 22. Key Vocabulary central angle minor arc major arc semicircle measure minor arc, major arc congruent circles congruent arcs central angle of a circle is an angle whose vertex is the center of the circle. In the diagram, is a central angle of (. If m is less than 1808, then the points on ( that lie in the interior of form a minor arc with endpoints and. he points on ( that do not lie on minor arc form a major arc with endpoints and. semicircle is an arc with endpoints that are the endpoints of a diameter. NMING Minor arcs are named by their endpoints. he minor arc associated with is named. Major arcs and semicircles are named by their endpoints and a point on the arc. he major arc associated with can be named. major arc $ minor KE ONE For our Notebook Measuring rcs he measure of a minor arc is the measure of its central angle. he expression m is read as the measure of arc. he measure of the entire circle is he measure of a major arc is the difference between 3608 and the measure of the related minor arc. he measure of a semicircle is m m E M L E 1 Find measures of arcs Find the measure of each arc of (, where } is a diameter. a. b. c. olution 1108 a. is a minor arc, so m 5 m b. is a major arc, so m c. } is a diameter, so is a semicircle, and m Find rc Measures 659

2 JEN wo arcs of the same circle are adjacent if they have a common endpoint. ou can add the measures of two adjacent arcs. OULE OULE 23 rc ddition ostulate he measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. For our Notebook m 5 m 1 m E M L E 2 Find measures of arcs UVE recent survey asked teenagers if they would rather meet a famous musician, athlete, actor, inventor, or other person. he results are shown in the circle graph. Find the indicated arc measures. a. m b. m c. m d. m E Whom Would ou ather Meet? Musician thlete Inventor 798 Other E ctor MEUE he measure of a minor arc is less than he measure of a major arc is greater than olution a. m 5 m 1 m b. m 5 m 1 m c. m m d. m E me GUIE IE for Examples 1 and 2 Identify the given arc as a major arc, minor arc, or semicircle, and find the measure of the arc. 1. Q 2. Q 3. Q 4. Q ONGUEN ILE N wo circles are congruent circles if they have the same radius. wo arcs are congruent arcs if they have the same measure and they are arcs of the same circle or of congruent circles. If ( is congruent to (, then you can write ( > (. 660 hapter 10 roperties of ircles

3 E M L E 3 Identify congruent arcs ell whether the red arcs are congruent. Explain why or why not. a. E b. c F U V olution a. > EF because they are in the same circle and m 5 mef. b. and U have the same measure, but are not congruent because they are arcs of circles that are not congruent. c. V > because they are in congruent circles and mv 5 m. at classzone.com GUIE IE for Example 3 ell whether the red arcs are congruent. Explain why or why not M N EEIE KILL IE HOMEWOK KE 5 WOKE-OU OLUION on p. W1 for Exs. 5, 13, and 23 5 NIE E IE Exs. 2, 11, 17, 18, and VOUL opy and complete: If and E are congruent central angles of (, then and E are?. 2. WIING What do you need to know about two circles to show that they are congruent? Explain. EMLE 1 and 2 on pp for Exs MEUING } and } E are diameters of (F. etermine whether the arc is a minor arc, a major arc, or a semicircle of (F. hen find the measure of the arc E E F E Find rc Measures 661

4 11. MULILE HOIE In the diagram, } Q is a diameter of (. Which arc represents a semicircle? Q Q Q Q EMLE 3 on p. 661 for Exs ONGUEN ell whether the red arcs are congruent. Explain why or why not L 858 M 14. V W N 15. EO NLI Explain what is wrong with the statement. ou cannot tell if ( > ( because the radii are not given. 16. wo diameters of ( are } and }. If m 5 208, find m and m. 17. MULILE HOIE ( has a radius of 3 and has a measure of 908. What is the length of }? 3Ï } 2 3Ï } HO EONE On (, mef , m FG 5, and m EFG If H is on ( so that m GH , explain why H must be on EF. 19. EONING In (, m 5 608, m 5 258, m 5 708, and me Find two possible values for me. 20. HLLENGE In the diagram shown, } Q }, }Q is tangent to (, and mv What is mu? U V 21. HLLENGE In the coordinate plane shown, is at the origin. Find the following arc measures on (. a. m y (3, 4) (4, 3) b. m c. m (5, 0) x WOKE-OU OLUION on p. W1 5 NIE E IE

5 OLEM OLVING EMLE 1 on p. 659 for Ex IGE he deck of a bascule bridge creates an arc when it is moved from the closed position to the open position. Find the measure of the arc. 23. On a regulation dartboard, the outermost circle is divided into twenty congruent sections. What is the measure of each arc in this circle? 24. EENE EONE surveillance camera is mounted on a corner of a building. It rotates clockwise and counterclockwise continuously between Wall and Wall at a rate of 108 per minute. a. What is the measure of the arc surveyed by the camera? b. How long does it take the camera to survey the entire area once? c. If the camera is at an angle of 858 from Wall while rotating counterclockwise, how long will it take for the camera to return to that same position? d. he camera is rotating counterclockwise and is 508 from Wall. Find the location of the camera after 15 minutes. 25. HLLENGE clock with hour and minute hands is set to 1:00.M. a. fter 20 minutes, what will be the measure of the minor arc formed by the hour and minute hands? b. t what time before 2:00.M., to the nearest minute, will the hour and minute hands form a diameter? MIE EVIEW EVIEW repare for Lesson 10.3 in Exs etermine if the lines with the given equations are parallel. (p. 180) 26. y 5 5x 1 2, y 5 5(1 2 x) 27. 2y 1 2x 5 5, y x 28. race n and point. raw a counterclockwise rotation of n 1458 about. (p. 598) Find the product. (p. 641) 29. (x 1 2)(x 1 3) 30. (2y 2 5)(y 1 7) 31. (x 1 6)(x 2 6) 32. (z 2 3) (3x 1 7)(5x 1 4) 34. (z 2 1)(z 2 4) E IE for Lesson 10.2, p. 914 ONLINE QUI at classzone.com 663