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11.3 Areas of Regular Polygons and 11.4 Use Geometric Probability

11.3 Areas of Regular Polygons and 11.4 Use Geometric Probability. You will find areas of regular polygons inscribed in circles. Essential Question: How do you find the area of a regular polygon?. You will learn how to answer this question by dividing the polygon into n isosceles triangles.

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11.3 Areas of Regular Polygons and 11.4 Use Geometric Probability

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  1. 11.3 Areas of Regular Polygonsand11.4 Use Geometric Probability • You will find areas of regular polygons inscribed in circles. Essential Question: • How do you find the area of a regular polygon? You will learn how to answer this question by dividing the polygon into n isosceles triangles. • How do you find the probability • that a point randomly selected in a • region is in a particular part of that • region? • You will use lengths and areas to find geometric probabilities. You will learn how to answer this question by comparing the measure of the part of the region to the measure of the entire region.

  2. In the diagram, ABCDEis a regular pentagon inscribed in F. Find each angle measure. 360° 5 a. m AFB AFB is a central angle,som AFB = , or 72°. Find angle measures in a regular polygon EXAMPLE 1 SOLUTION

  3. In the diagram, ABCDEis a regular pentagon inscribed in F. Find each angle measure. b. m AFG FG is an apothem, which makes it an altitude ofisosceles ∆AFB. So, FGbisectsAFB andm AFG = m AFB = 36°. 1 2 Find angle measures in a regular polygon EXAMPLE 1 SOLUTION

  4. In the diagram, ABCDEis a regular pentagon inscribed in F. Find each angle measure. c. m GAF The sum of the measures of right ∆GAF is 180°. So, 90° + 36° + m GAF = 180°, andm GAF = 54°. Find angle measures in a regular polygon EXAMPLE 1 SOLUTION

  5. In the diagram, WXYZis a square inscribed in P. 1. Identify the center, a radius, an apothem, and a central angle of the polygon. ANSWER P, PY or XP, PQ, XPY. for Example 1 GUIDED PRACTICE

  6. 2. Find m XPY, m XPQ, andm PXQ. ANSWER 90°, 45°, 45° for Example 1 GUIDED PRACTICE

  7. You are decorating the top of a table by covering it with small ceramic tiles. The table top is a regular octagon with 15inch sides and a radius of about 19.6inches. What is the area you are covering? Find the area of a regular polygon EXAMPLE 2 DECORATING SOLUTION STEP 1 Find the perimeter Pof the table top. An octagon has 8 sides, so P = 8(15) = 120inches.

  8. Find the apothem a. The apothem is height RSof ∆PQR. Because ∆PQRis isosceles, altitude RSbisects QP. 1 1 So,QS = (QP) = (15) = 7.5 inches. 2 2 √ a = RS ≈ √19.62 – 7.52 = 327.91 ≈ 18.108 Find the area of a regular polygon EXAMPLE 2 STEP 2 To find RS, use the Pythagorean Theorem for ∆ RQS.

  9. 1 A = aP 2 1 ≈ (18.108)(120) 2 ANSWER So, the area you are covering with tiles is about 1086.5square inches. Find the area of a regular polygon EXAMPLE 2 STEP 3 Find the area Aof the table top. Formula for area of regular polygon Substitute. ≈ 1086.5 Simplify.

  10. 3. ANSWER about 46.6 units, about 151.5 units2 for Examples 2 and 3 GUIDED PRACTICE Find the perimeter and the area of the regular polygon.

  11. • The center and radius of a regular polygon are the center and radius of its circumscribed circle. • The distance from the center to a side of a regular polygon is the apothem. • A central angle of a regular polygon is formed by two consecutive radii. • The area of a regular polygon is A =(1/2) where a is the apothem and P is the perimeter. • You will find areas of regular polygons inscribed in circles. Essential Question: • How do you find the area of a regular polygon? Find the length a of the apothem and the perimeter P of the polygon. Substitute those values into the formula for the area of a regular polygon A =(1/2)aP

  12. Find the probability that a point chosen at random on PQis on RS. – – 6 3 Length of RS Length of PQ 4 ( 2) 5 ( 5) , = = P(Point is on RS)= = – – 10 5 EXAMPLE 1 Use lengths to find a geometric probability SOLUTION 0.6, or 60%.

  13. EXAMPLE 2 Use a segment to model a real-world probability MONORAIL A monorail runs every 12 minutes. The ride from the station near your home to the station near your work takes 9 minutes. One morning, you arrive at the station near your home at 8:46. You want to get to the station near your work by 8:58. What is the probability you will get there by 8:58?

  14. EXAMPLE 2 Use a segment to model a real-world probability SOLUTION STEP 1 Find: the longest you can wait for the monorail and still get to the station near your work by 8:58. The ride takes 9 minutes, so you need to catch the monorail no later than 9 minutes before 8:58, or by 8:49. The longest you can wait is 3 minutes (8:49 – 8:46 = 3 min).

  15. Model the situation. The monorail runs every 12 minutes, so it will arrive in 12 minutes or less. You need it to arrive within 3 minutes. EXAMPLE 2 Use a segment to model a real-world probability STEP 2 The monorail needs to arrive within the first 3 minutes.

  16. P(you get to the station by 8:58) Favorable waiting time 3 1 = = = Maximum waiting time 12 4 The probability that you will get to the station by 8:58. is 1 ANSWER or 25%. 4 EXAMPLE 2 Use a segment to model a real-world probability STEP 3 Find: the probability.

  17. Find the probability that a point chosen at random on PQis on the given segment. Express your answer as a fraction, a decimal, and a percent. RT 1. 1 ANSWER , 0.1, 10% 10 for Examples 1 and 2 GUIDED PRACTICE

  18. TS Find the probability that a point chosen at random on PQis on the given segment. Express your answer as a fraction, a decimal, and a percent. 2. ANSWER 1 , 0.5, 50% 2 for Examples 1 and 2 GUIDED PRACTICE

  19. Find the probability that a point chosen at random on PQis on the given segment. Express your answer as a fraction, a decimal, and a percent. PT 3. ANSWER 2 , 0.4, 40% 5 for Examples 1 and 2 GUIDED PRACTICE

  20. Find the probability that a point chosen at random on PQis on the given segment. Express your answer as a fraction, a decimal, and a percent. RQ 4. ANSWER 7 , 0.7, 70% 10 for Examples 1 and 2 GUIDED PRACTICE

  21. ANSWER 1 or 50%. 2 for Examples 1 and 2 GUIDED PRACTICE 5. WHAT IF?In Example 2, suppose you arrive at the station near your home at 8:43. What is the probability that you will get to the station near your work by 8:58?

  22. ARCHERY EXAMPLE 3 Use areas to find a geometric probability The diameter of the target shown at the right is 80 centimeters. The diameter of the red circle on the target is 16 centimeters. An arrow is shot and hits the target. If the arrow is equally likely to land on any point on the target, what is the probability that it lands in the red circle?

  23. Area of red circle P(arrow lands in red region) = Area of target (82) 64 1 = = = (402) 1600 25 ANSWER The probability that the arrow lands in the red region is 1 , or 4%. 25 EXAMPLE 3 Use areas to find a geometric probability SOLUTION Find the ratio of the area of the red circle to the area of the target.

  24. SCALE DRAWING Find the area of the field. The shape is a rectangle, so the area is bh = 10 3 = 30 square units. EXAMPLE 4 Estimate area on a grid to find a probability Your dog dropped a ball in a park. A scale drawing of the park is shown. If the ball is equally likely to be anywhere in the park, estimate the probability that it is in the field. SOLUTION STEP 1

  25. Find the total area of the park. EXAMPLE 4 Estimate area on a grid to find a probability STEP 2 Count the squares that are fully covered. There are 30 squares in the field and 22 in the woods. So, there are 52 full squares. Make groups of partially covered squares so the combined area of each group is about 1 square unit. The total area of the partial squares is about 6 or 7 square units. So, use 52 + 6.5 =58.5 square units for the total area.

  26. Area of field 300 30 20 P(ball in field) = = = Total area of park 58.5 39 585 The probability that the ball is in the field is about 20 , or 51.3%. ANSWER 39 EXAMPLE 4 Estimate area on a grid to find a probability STEP 3 Write a ratio of the areas to find the probability.

  27. 14 ANSWER 25 = 56% for Examples 3 and 4 GUIDED PRACTICE 6. In the target in Example 3, each ring is 8 centimeters wide. Find the probability that an arrow lands in a black region.

  28. 19 or about 48.7%. ANSWER 39 for Examples 3 and 4 GUIDED PRACTICE 7. In Example 4, estimate the probability that the ball is in the woods.

  29. How do you find the probability • that a point randomly selected in a • region is in a particular part of that • region? • You will use lengths and areas to find geometric probabilities. • The probability of an event is a measure of the likelihood that an event will occur. It is a number between 0 and 1, inclusive. • Geometric probability is a ratio involving lengths or areas. Find the ratio of the measure of the specific part of the region to the measure of the entire region.

  30. ANSWER 15° 2. ANSWER 110 cm2 Daily Homework Quiz 1. Find the measure of the central angle of a regular polygon with 24 sides. Find the area of each regular polygon.

  31. 3. ANSWER 374.1 cm2 Daily Homework Quiz

  32. 4. ANSWER 99.4 in. ; 745.6 in.2 Daily Homework Quiz Find the perimeter and area of each regular polygon.

  33. 5. 22.6 m; 32 m2 ANSWER Daily Homework Quiz

  34. 1. Find the probability that a point chosen at random on JN is on KN. ANSWER 5 or 71.4% 7 Daily Homework Quiz

  35. 2. Find the probability that a randomly chosen point in the figure will lie in the shaded region. ANSWER 36.3% Daily Homework Quiz

  36. 3. A bird is equally likely to be anywhere in the garden shown. What is the probability that the bird will be in the vegetables? ANSWER 24 aboutor 47% 51 Daily Homework Quiz

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