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FeatureLesson Geometry Lesson Main 1. A park contains two circular playgrounds. One has a diameter of 60 m, and the other has a diameter of 40 m. How much.

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Presentation on theme: "FeatureLesson Geometry Lesson Main 1. A park contains two circular playgrounds. One has a diameter of 60 m, and the other has a diameter of 40 m. How much."— Presentation transcript:

1 FeatureLesson Geometry Lesson Main 1. A park contains two circular playgrounds. One has a diameter of 60 m, and the other has a diameter of 40 m. How much greater is the area of the larger playground? Round to the nearest whole number. 2. A circle has an 8-in. radius. Find the area of a sector whose arc measures 135. Leave your answer in terms of. For Exercises 3 and 4, find the area of the shaded segment. Round to the nearest whole unit m 2 24 in cm in. 2 Lesson 10-7 Areas of Circles and Sectors Lesson Quiz 10-8

2 FeatureLesson Geometry Lesson Main Lesson 10-8 Geometric Probability (For help, go to the Skills Handbook, pages 756 and 762.) Two circles have radii 1 m and 2 m, respectively. What is the simplest form of the fraction with numerator equal to the area of the smaller circle and denominator equal to the area of the larger circle? an odd number 7. 2 or 58. a prime number BD AE AB BC CE AF Find and simplify each ratio. You roll a number cube. Find the probability of rolling each of the following. Check Skills Youll Need 10-8 The area of the smaller circle is r2 = (1)2 = m; The area of the larger circle is r 2 = (2) 2 = 4 m; smaller : larger = : 4 = 1 : 4, or ¼.

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6 FeatureLesson Geometry Lesson Main Lesson 10-8 Geometric Probability Notes 10-8 You may recall that the probability of an event is the ratio of the number of favorable outcomes to the number of possible outcomes.

7 FeatureLesson Geometry Lesson Main Lesson 10-8 Geometric Probability Notes 10-8 Geometric probability is used when an experiment has an infinite number of outcomes. In geometric probability, the probability of an event is based on a ratio of geometric measures such as length or area. The outcomes of an experiment may be points on a segment or in a plane figure.

8 FeatureLesson Geometry Lesson Main Lesson 10-8 Geometric Probability Notes 10-8 For example, if points of segments represent outcomes, then

9 FeatureLesson Geometry Lesson Main Lesson 10-8 Geometric Probability Notes 10-8 If an event has a probability p of occurring, the probability of the event not occurring is 1 – p. Remember!

10 FeatureLesson Geometry Lesson Main The length of the segment between 2 and 10 is 10 – 2 = 8. The length of the ruler is 12. P(landing between 2 and 10) = =, or length of favorable segment length of entire segment A gnat lands at random on the edge of the ruler below. Find the probability that the gnat lands on a point between 2 and 10. Lesson 10-8 Geometric Probability Quick Check Additional Examples 10-8 Finding Probability Using Segments

11 FeatureLesson Geometry Lesson Main A museum offers a tour every hour. If Benny arrives at the tour site at a random time, what is the probability that he will have to wait at least 15 minutes? Because the favorable time is given in minutes, write 1 hour as 60 minutes. Benny may have to wait anywhere between 0 minutes and 60 minutes. Starting at 60 minutes, go back 15 minutes. The segment of length 45 represents Bennys waiting more than 15 minutes. P(waiting more than 15 minutes) =, or Represent this using a segment. The probability that Benny will have to wait at least 15 minutes is, or 75% Lesson 10-8 Geometric Probability Quick Check Additional Examples 10-8 Real-World Connection

12 FeatureLesson Geometry Lesson Main Lesson 10-8 Geometric Probability Notes 10-8 If the points of a region represent equally-likely outcomes, then you can find probabilities by comparing areas.

13 FeatureLesson Geometry Lesson Main Find the area of the square. A = s 2 = 20 2 = 400 cm 2 Find the area of the circle. Because the square has sides of length 20 cm, the circles diameter is 20 cm, so its radius is 10 cm. A = r 2 = (10) 2 = 100 cm 2 Find the area of the region between the square and the circle. A = (400 – 100 ) cm 2 A circle is inscribed in a square target with 20-cm sides. Find the probability that a dart landing randomly within the square does not land within the circle. Lesson 10-8 Geometric Probability Additional Examples 10-8 Finding Probability Using Area

14 FeatureLesson Geometry Lesson Main (continued) Use areas to calculate the probability that a dart landing randomly in the square does not land within the circle. Use a calculator. Round to the nearest thousandth. The probability that a dart landing randomly in the square does not land within the circle is about 21.5%. P (between square and circle) = = = 1 – area between square and circle area of square 400 – Lesson 10-8 Geometric Probability Quick Check Additional Examples 10-8

15 FeatureLesson Geometry Lesson Main To win a prize, you must toss a quarter so that it lands entirely within the outer region of the circle below. Find the probability that this happens with a quarter of radius in. Assume that the quarter is equally likely to land anywhere completely inside the large circle The center of a quarter with a radius of in. must land at least in. beyond the boundary of the inner circle in order to lie entirely outside the inner circle. Because the inner circle has a radius of 9 in., the quarter must land outside the circle whose radius is 9 in. + in., or 9 in Lesson 10-8 Geometric Probability Additional Examples 10-8 Real-World Connection

16 FeatureLesson Geometry Lesson Main (continued) Similarly, the center of a quarter with a radius of in. must land at least in. within the outer circle. Because the outer circle has a radius of 12 in., the quarter must land inside the circle whose radius is 12 in. – in., or 11 in Lesson 10-8 Geometric Probability Find the area of the circle with a radius of 9 in. A = r 2 = (9 ) in Find the area of the circle with a radius of 11 in. A = r 2 = (11 ) in Additional Examples 10-8

17 FeatureLesson Geometry Lesson Main (continued) Use the area of the outer region to find the probability that the quarter lands entirely within the outer region of the circle. The probability that the quarter lands entirely within the outer region of the circle is about 0.326, or 32.6%. Lesson 10-8 Geometric Probability P (outer region) = area of outer region area of large circle – = Quick Check Additional Examples 10-8

18 FeatureLesson Geometry Lesson Main 1. A point on AF is chosen at random. What is the probability that it is a point on BE? 2. Express elevators to the top of a tall building leave the ground floor every 40 seconds. What is the probability that a person would have to wait more than 30 seconds for an express elevator? A dart you throw is equally likely to land at any point on each board shown. For Exercises 3–5, find the probability of its landing in the shaded area. 3. regular octagon4. square5. circle , or 37.5% 1212, or 50% 8 25, or 32% Lesson 10-8 Geometric Probability Lesson Quiz 10-8

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