Geometry Semester 2 Mrs. Day-Blattner Period 2 Agenda March 28, 2017 1) Bulletin 2) Spring Break Work Page 85 - 87 Questions 2 through 15 - self-check (answers posted in homework assignment itself - click on homework tab and past due assignments) 3) Radians and Arc Length (page 89 -) reading - highlighting 4) Determine the Arc Length pg 91 5) Area of a Sector starting pg 93 6) Homework Remember: What is a radian? The measure of the central angle that subtends an arc length of one radius. Remember: Circumference = 2 r And so 2 radians = 360° Notes Lesson 12. Radians and Arc Length p 89 Read out loud - one person 1st paragraph Go over equations together 2nd person - reads from “Now if…” to end of page. To do: Highlight definition of arc length. What symbol is used for arc length? What might be a more helpful phrase than “percentage of the circumference” for arc length? Notes Lesson 12. Radians and Arc Length p 89 The arc length of an arc is the physical distance along the circumference between the two points of the arc. What symbol is used for arc length? What might be a more helpful phrase than “percentage of the circumference” for arc length? Notes Lesson 12. Radians and Arc Length p 89 The arc length of an arc is the physical distance along the circumference between the two points of the arc. What symbol is used for arc length? Lowercase s What might be a more helpful phrase than “percentage of the circumference” for arc length? Notes Lesson 12. Radians and Arc Length p 89 So to calculate a portion of that circumference we calculate a percentage of the circumference. Notes Lesson 12. Radians and Arc Length p 89 So to calculate a portion of that Fraction circumference we calculate a percentage of the circumference. Percentage - number or amount in each hundred (out of a hundred) Arc length, s = x° 2 r 360° Page 90. Example #1 What is the length of an arc if the circle has a radius of 4cm and the central angle is 120° Arc length = 120° 2 (4cm) = 8 cm 3 360° s approx 8.37 cm Arc length, s = x° 2 r 360° s= x rad 2 r =xr 2 rad Page 90. Example #2 What is the length of an arc if the circle has a radius of 10cm and the central angle is 4 /5 radians? Arc length = 4 (10cm) = 5 Arc length, s = x° 2 r 360° s= x rad2 r = xr 2 rad Page 90. Example #2 What is the length of an arc if the circle has a radius of 10cm and the central angle is 4 /5 radians? Arc length = 4 (10cm) = 8 cm Exact 5 s approx 25.13 cm Arc length, s = θ° 2 r 360° s= θ rad 2 r =θr 2 rad Arc length, s = θ° 2 r s = θ rad 2 r =θr 360° 2 Page 91. 1. Determine the arc length. a) central angle is 30°, radius of 3cm Arc length, s = 30° 2 (3cm) = cm 2 360° (E - for exact, leave in terms of pi) Arc length x° 2 r 360° Page 91. 1. Determine the arc length. b) central angle is 90°, radius of 8cm Arc length, s = 90° 2 (8cm) = 4 cm 360° (E - for exact, leave in terms of pi) Arc length x° 2 r 360° Page 91. 1. Determine the arc length. c) central angle is 72°, radius of 10cm Arc length, s = 72° 2 (10cm) = 4 cm 360° (E - for exact, leave in terms of pi) Arc length, s = θ° 2 r s = θ rad 2 r =θr 360° 2 Page 91. 1. Determine the arc length. Complete remaining problems on page 91 and 92 d) central angle is /4 rad, radius of 12cm Arc length, s = (12cm) = 3 cm 4 (E - for exact, leave in terms of pi) Arc length, s = θ° 2 r s = θ rad 2 r =θr 360° 2 Page 91. 1. Determine the arc length. Complete remaining problems on page 91 and 92 e) central angle is 2 /3 rad, radius of 15cm Arc length, s = 2 (15cm) = 10 cm 3 (E - for exact, leave in terms of pi) Arc length, s = θ° 2 r s = θ rad 2 r =θr 360° 2 Page 91. 1. Determine the arc length. Complete remaining problems on page 91 and 92 f) central angle is 4 /5 rad, radius of 10cm Arc length, s = 4 (10cm) = 8 cm 5 (E - for exact, leave in terms of pi) 2. Arc length is a fraction of the circumference, 2 r. When we use radians to measure the central angle, θ, we get: s= θ rad 2 r =θr 2 See how the 2 terms cancel each other out? 3. On white board - one pair from each set of tables. s = 4 (10cm) = 8 cm 5 b) s = 7 (3cm) = 7 cm 2 6 c) s = 3 (4cm) = 6 cm 2 d) s = (18cm) = 3 cm 6 a) 4. Circle G has a radius of 7cm. After computing an arc on the circle G, Nancy finds the arc length to be 14cm. How did she know the central angle must be 2 radians? s = θr 4. Circle G has a radius of 7cm. After computing an arc on the circle G, Nancy finds the arc length to be 14cm. How did she know the central angle must be 2 radians? s =θr 14cm = θ7cm θ = 14cm/7cm = 2 When we measure angles in radians. So central angle must be 2 radians. 5. Determine the missing information. a)s = 4 cm r = 8cm s =θr 4 cm = θ8cm θ = 4 cm/8cm = ½ radians 5. Determine the missing information. b)s = 8cm θ = 0.8rad s =θr 8cm = 0.8rcm r= 8cm/0.8rad = 10cm r = 10cm 5. Determine the missing information. c)r = 4.5cm θ = /3 rad s =θr s= 1.5 cm s = /3 (4.5cm) 5. Determine the missing information. d)s = 28 cm θ = 7 /4 rad s =θr 28 cm = 7 /4 (r) r= 4(28) cm 7 r = 16cm 5. Determine the missing information. e)s = 10 cm r = 8cm s =θr 10 cm =θ 8cm θ = 10 cm 8cm θ = 5 radians 4 5. Determine the missing information. f)θ = 2 rads s = 5 cm 5 s =θr 5 cm =2 r 5 r = 5(5 cm) 2 r = 12.5 cm 5. Determine the missing information. g)r = 8 cm θ = /2 rad s =θr s= 4 cm s = /2 (8cm) 5. Determine the missing information. h)θ = 5 rads s = 10 cm 6 s =θr 10 cm =5 r 2 r = 6(10 cm) 5 r = 12 cm 6. Find the radius of the circle in which a central angle of 5 radians intercepts an arc of 62.5 feet. θ = 5 rads s =θr s = 62.5 feet 62.5 feet =5 r r = 62.5 feet 5 r = 12.5 feet 7. Find the measure (in radians) of a central angle that intercepts an arc of length 16cm in a circle with radius of 8cm. r= 8cm s =θr θ = 16cm 8cm s = 16cm 16cm =8cmθ θ = 2 rad 7. Find the measure (in radians) of a central angle that intercepts an arc of length 24 cm in a circle with radius of 10cm. r= 10cm s =θr θ = 24 cm 10cm s = 24 cm 24 cm =10cmθ θ = 2.4 rad (or 12/5 rad) Notes Lesson 13. Area of a sector. Sector: Let AB be an arc of a circle with center O and radius r. The union of A line segments OP, where P is any point of AB, is called a sector. B r O A x° Sector: Let AB be an arc of a circle with center O and radius r. The union of A line segments OP, where P is any point of AB, is called a sector. B r O P A Sector: Let AB be an arc of a circle with center O and radius r. The union of A line segments OP, where P is any point of AB, is called a sector. B r O P A Sector: Think of a sector as B fraction a part or a percentage of the whole area of the circle. r O P Area = A Sector: Think of a sector as B fraction a part or a percentage of the whole area of the circle. Area = r2 r O P A Area of sector = θ° r2 360° OR θ 2 r2 = θr2 2 Don’t use half of one and half of the other! Area of sector = θ° r2 360° OR θ 2 r2 = θr2 2 Page 95. 1. Determine the area of the sector. a) r = 8cm θ = /4 rad A= . 1(8 cm)2 = 64 cm2 = 8 8 4 2 (E - for exact, leave in terms of Be careful multiplying 2 fractions of pi cm and remember to square radius before trying to find common pi) factors Area of sector = θ° r2 360° OR θ 2 r2 = θr2 2 Page 95. 1. Determine the area of the sector. b) r = 3cm θ = 5 /3 rad A = 5 . 1(3 cm)2 = 5(9) cm2 = 45 cm2 6 3 2 6 A = 15 cm2 2 θ r2 360° Page 95. 1. Determine the area of the sector. Area of sector = c) θ,central angle, is 60°, d= 4cm, r = 2cm A = 60° (2cm)2 = 2 cm2 3 360° (E - for exact, leave in terms of pi) Area of sector = θ° r2 360° OR θ 2 r2 = θr2 2 Page 95. 1. Determine the area of the sector. d) r = 12cm θ = 3 /2 rad A = 3 . 1(12 cm)2 = 3(144) cm2 = 3(36) cm2 2 2 4 A = 108 cm2 θ r2 360° Page 95. 1. Determine the area of the sector. Area of sector = e) θ,central angle, is 240°, d= 6cm, r = 3cm A = 240° (3cm)2 = 6 cm2 360° (E - for exact, leave in terms of pi) Area of sector = θ° r2 360° OR θ 2 r2 = θr2 2 Page 95. 1. Determine the area of the sector. f) d = 10cm θ = 2 /5 rad r = 5cm A = 2 . 1(5cm)2 = (25) cm2 = 5 cm2 5 2 5 A = 5 cm2 2. Prepare a good explanation for yourself and for our absentee students to share with them on Thursday. Area of sector = θ° r2 360° OR θ 2 r2 = θr2 2 3. Determine the required information. (looking for sector with unbroken arc line, unshaded) a) r = 10cm θ = 2 rad r = 5cm A = 2 . 1(10cm)2 = (100) cm2 = 100cm2 2 Area of sector = θ° r2 360° OR θ 2 r2 = θr2 2 3. Determine the required information. (looking for sector with unbroken arc line, unshaded) b) A = 6 cm2 θ = /3 rad 6 cm2 = . 1(rcm)2 = (rcm)2 3 2 6 6 (6 cm2) = (rcm)2 3. Determine the required information. (looking for sector with unbroken arc line, unshaded) b) A = 6 cm2 θ = /3 rad 6 cm2 = . 1(rcm)2 = (rcm)2 3 2 6 2 2 6 (6 cm ) = (rcm) r = √36cm2 r = 6cm θ r2 360° Page 95. 3. Determine the required information. Area of sector = c) θ,central angle, is 300°, r = 3cm (looking for sector with unbroken arc line, unshaded) A = 300° 360° (3cm)2 = 45 cm2 = 7.5 cm2 6 (E - for exact, leave in terms of pi) Homework. Heads up Circles Test on Tuesday, April 4th Finish Lesson 12 Arc Length worksheet problems, page 91 and 92 Finish Lesson 13 Area of a Sector Worksheet problems, page 95 and 96 Learn Vocabulary in Lesson Summary Box pg 101 (vocab quiz on Thursday) Write the steps you would need to take to solve the 3 problems on page 118 (research unfamiliar terms). Page 95. 3. Determine the required information. d) r = 6cm s = 5 cm s= θr so 5 cm = θ(6cm) θ= 5 cm 6cm = 5 radians 6 (Use extra space below the question to complete) Page 95. 3. Determine the required information. d) r = 6cm s = 5 cm s= θr so 5 cm = θ(6cm) θ= 5 cm 6cm = 5 radians 6 Area = ½ θ r2 = ½ 5 (6cm)2 = 5 36cm2 = 15 cm2 6 12 Page 95. 3. Determine the required information. e) r = 4cm θ = /2 rad Area = ½ θ r2 = ½ (4cm)2 = 16cm2 = 4 cm2 4 2 Page 95. 3. Determine the required information. f) A = 33 cm2 r = 6cm Area = ½ θ r2 = 33 cm2 =½ θ(6cm)2 = θ(36cm2) 2 2 2(33 cm ) = θ 36cm2 Extra notes: 33/3 = 11 36/2 =18 18/3 = 6 θ = 11 6cm2

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