9.2 Angles and Arcs STANDARDS Objectives Content: G.C.2, G.C.5 Practices: 1, 2, 3, 4, 6, 7, 8 Use with Lesson 10-2 • Derive and apply the formula for arc length. • Derive the fact that the length of the arc intercepted by an angle is proportional to the radius. • Define and apply radian measure. A central angle of a circle is an angle in which the vertex is the center of the circle. In the figure, ∠ACB is a central angle. An arc is a portion of a circle defined by two endpoints. A minor arc is the shortest arc connecting two endpoints. A major arc is the longest arc connecting two endpoints. minor arc AB B A C major arc ADB Arc length is the distance between endpoints along an arc. D EXAMPLE 1 Investigate Arc Length G.C.2 EXPLORE An architect is designing the seating area for a theater. The seating area is formed by a region that lies between two circles, as shown in the figure. The architect is planning to place a brass rail in front of the first row of seats. She wants to know the length of the rail. a. COMMUNICATE PRECISELY In the figure, the arc that represents the brass rail is marked 120°. Explain what this means. SMP 6 seating area 120° R S T 15 ft brass rail b. CALCULATE ACCURATELY Explain how to find the circumference of T. Express the circumference in terms of π. SMP 6 c. REASON ABSTRACTLY Explain how you can use your answer to part b and proportional reasoning to find the length of the brass rail. Express the length in terms of π and to the nearest tenth of a foot. SMP 2 294 CHAPTER 9 Circles Copyright © McGraw-Hill Education d. DESCRIBE A METHOD The ⏜architect is considering changing the radius of T or RS. Describe a general method she can use to find the length changing the measure of ⏜ of RS . SMP 8 e. USE A MODEL The seating area is 15 feet deep, so the distance from the center of the circle to the edge of the seating area is 30 feet. If the architect wants to surround the entire seating area with a brass rail, what length of rail does she need? Explain your reasoning. SMP 4 The Key Concept box summarizes the relationship you discovered in the previous exploration. KEY CONCEPT Arc Length Complete the proportion and the equation. The ratio of the length of an arc to the circumference of the circle is equal to the ratio of the degree measure of the arc to 360. 2π r Proportion: ______ = __________ EXAMPLE 2 x° r Equation: = Apply Arc Length G.C.2 A model train is set up on a circular track with a diameter of 8 feet. As the train travels from the station to the coal mine, it moves 84° around the track. The train moves at a constant rate of 6 inches per second. Follow these steps to determine how long it takes the train to travel from the station to the coal mine. a. INTERPRET PROBLEMS Sketch and label a figure that represents this situation in the space at the right. SMP 1 b. CALCULATE ACCURATELY Explain how to find the length of the track from the station to the coal mine to the nearest hundredth of a foot. SMP 6 Copyright © McGraw-Hill Education c. CALCULATE ACCURATELY Explain how to find the time it takes the train to travel from the station to the coal mine to the nearest second. SMP 6 d. CRITIQUE REASONING A student said that if the radius of the track were doubled, then the amount of time it takes the train to travel from the station to the coal mine would also be doubled. Do you agree? Explain. SMP 3 9.2 Angles and Arcs 295 e. CRITIQUE REASONING A student said that if the angle were doubled, then the amount of time it takes the train to travel from the station to the coal mine would also be doubled. Do you agree? Explain. SMP 3 EXAMPLE 3 Investigate Proportionality G.C.2, G.C.5 Follow these steps to investigate how the length of the arc intercepted by a central angle of a circle is related to the radius. 45° 3 a. CALCULATE ACCURATELY The figure shows an arc intercepted by a central angle of 45°. Show how to find the length of the arc in terms of π. SMP 1 b. FIND A PATTERN Enlarge the circle so the measure of the central angle remains 45. Complete the table by finding the length of the arc (in terms of π) for each of the given radii. SMP 7 Radius of circle, r 3 5 11 15 r Length of arc, c. COMMUNICATE PRECISELY Look for a pattern. When the measure of the central angle is 45, what type of relationship do you notice between the arc length and the radius? Explain. SMP 6 d. FIND A PATTERN Repeat the above process and complete the following table, but this time consider arcs with a central angle of 60°. SMP 7 Radius of circle, r 2 4 5 10 r 60° r Length of arc, f. DESCRIBE A METHOD Now generalize your findings. Suppose the measure of the central angle is x. What type of relationship will exist between the arc length and the radius? How can you find the length of the arc if you know the radius? SMP 8 296 CHAPTER 9 Circles Copyright © McGraw-Hill Education e. COMMUNICATE PRECISELY Look for a pattern in your table. When the measure of the central angle is 60, what type of relationship do you notice between the arc length and the radius? Explain. SMP 6 You can use your findings from Example 3 to define the radian measure of an angle. Much like a foot and an inch are two different units for measuring the length of a line, a radian and a degree are two different units for measuring the size of an angle. An angle that measures 1 radian will mark off an arc on a circle that is equal in length to the radius of the circle. Because degrees and radians are two units of measurement, there is a conversion factor between them. KEY CONCEPT Radian Measure Complete the formula. The length of the arc intercepted by an angle is proportional to the radius. The constant of proportionality for this relationship is the radian measure of the angle. x° Formula: The radian measure of an angle of x°= EXAMPLE 4 Apply Degrees and Radians A carpenter is ordering metal brackets from a specialty company. The brackets are available in a variety of angles, as shown. The carpenter wants to know which of the brackets has the greatest angle measure and which of the brackets, if any, are a right angle. a. CALCULATE ACCURATELY Explain how to express the measure of Bracket A in radians. SMP 6 r G.C.5 Bracket Angle A 70° B 3π _____ radians 4 C 1 radian D π ___ radians 2 70° Bracket A Copyright © McGraw-Hill Education b. REASON QUANTITATIVELY Without doing any further calculations, can the carpenter determine which of the brackets has the greatest angle measure? Explain. SMP 2 c. DESCRIBE A METHOD Suppose you know the radian measure of an angle. Explain how you can convert the measure to degrees. SMP 8 d. CALCULATE ACCURATELY Are any of the brackets a right angle? Explain. SMP 6 e. CRITIQUE REASONING The carpenter orders a bracket that can be adjusted to any angle measure from 0 radians to π radians. The carpenter claims that this bracket can be adjusted to form a straight angle. Do you agree? Explain. SMP 6 9.2 Angles and Arcs 297 PRACTICE 1. The figure ⏜ shows a circular flower bed. Part of the fence surrounding the flower bed, AB, is damaged and needs to be replaced. The fencing costs $8.75 per linear foot. G.C.2 A 72° B a. USE A MODEL What will it cost to replace the damaged portion of the fence? SMP 4 18 ft b. REASON QUANTITATIVELY What is the length of the undamaged portion of the fence to the nearest tenth of a foot? SMP 2 2. COMMUNICATE PRECISELY The clock at the Palace of Westminster in London is best known by the name Big Ben. The minute hand of the clock is 14 feet long. Explain how you can determine how far the tip of the minute hand moves between 3 pm and 3:10 pm. Round to the nearest tenth. G.C.2, SMP 6 Q 3. At an amusement park, go-karts travel around a circular track with a radius of 90 feet. As the go-karts travel from point P to point Q, they cover a distance of 165 feet. G.C.2 a. USE A MODEL What is the measure of ∠QRP to the nearest degree? SMP 4 165 ft R 90 ft P b. DESCRIBE A METHOD Suppose you know the distance in feet, d, that a go-kart travels. How can you find the measure x (in degrees) of the arc that describes the go-kart’s path? Include a written and algebraic description. SMP 8 CALCULATE ACCURATELY Convert each degree measure to the equivalent measure in radians. G.C.5, SMP 6 5. 135° 298 CHAPTER 9 Circles 6. 18° 7. 65° 8. 120° Copyright © McGraw-Hill Education ⏜ ⏜ 4. CRITIQUE REASONING In C, AB has a central angle⏜ of 25°. In D, EF has a central angle of 25°. ⏜ A student claimed that the arc length of AB must be equal to the arc length of EF. Do you agree? If so, explain why. If not, explain whether there is ever a situation in which you can conclude that the arc lengths are equal. G.C.2, SMP 3 CALCULATE ACCURATELY Convert each radian measure to the equivalent measure in degrees. G.C.5, SMP 6 π 9. ___ radians 6 5π 10. _____ radians π 11. ____ radians 6 12 12. 3 radians REASON ABSTRACTLY Determine whether each statement is always, sometimes, or never true. Explain. G.C.5, SMP 2 13. If a central angle measures 2 radians, then the length of the arc it intercepts is twice the length of the circle’s radius. π 2 14. The radian measure of an obtuse angle is less than ___. 15. In a circle, the length of the arc intercepted by a central angle is equal to the radius of the circle. 16. An angle’s measure is 1 in radians and 45 in degrees. 17. USE A MODEL As a pendulum swings from point A to point B, it sweeps out an angle 5π of _____ radians. What is the distance the weight at the end of the pendulum travels as 18 it swings from A to B and back to A? Round to the nearest tenth. G.C.5, SMP 4 10 cm A 18. CRTIQUE REASONING An archery target consists ⏜ of three concentric⏜ circles, as shown. Latanya says that ST must be 3 times as long as QR. Melanie says that it is not possible to determine this relationship without knowing the measure of ∠SPT. Who is correct? Explain. G.C.2, G.C.5, SMP 3 B S Q P Copyright © McGraw-Hill Education 1 ft R 1 ft 1 ft T 19. REASON ABSTRACTLY A bicycle wheel has a diameter of 27.5 inches. On a trip through town, the wheel rotated 2143 times around plus an additional 62°. How far G.C.5, SMP 2 did the bicycle travel? 9.2 Angles and Arcs 299
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