Ex 6.3, 5

Transcript

Ex 6.4, 5 A chord of a circle of radius 15 cm subtends an angle of 60^∘ at the centre of the circle. Find the areas of the corresponding minor and major segments of the circle. (Use 𝜋≈3.14 and √3≈1.73.) Let the minor segment be APB Given that Radius = r = 15 cm 𝜃 = 60° Now, Area of segment APB = Area of sector OAPB – Area of ΔOAB Let’s find separately Area of sector OAPB Area of sector OAPB = 𝜃/360×𝜋𝑟2 = 𝟔𝟎/𝟑𝟔𝟎 × 𝟑.𝟏𝟒 × 𝟏𝟓 × 𝟏𝟓 = 1/6 × 3.14 × 15 × 15 = 1/2 × 3.14 × 5 × 15 = 1.57 × 5 × 15 = 117.75 cm2 Area of Δ AOB In ∆ AOB, OA = OB = 15 cm ∠ AOB = 60° We know that Angles opposite equal side are equal ∴ ∠ OAB = ∠ OBA By Angle sum property ∠ OAB + ∠ AOB + ∠ OBA = 180° ∠ OAB + 60° + ∠ OBA = 180° Putting ∠ OAB = ∠ OBA ∠ OAB + 60° + ∠ OAB = 180° 2 × ∠ OAB + 60° = 180° 2 × ∠ OAB = 180° – 60° 2 × ∠ OAB = 120° ∠ OAB = (120°)/2 ∠ OAB = 60° Thus, ∠ OBA = ∠ OAB = 60° So, all 3 angles are 60° Which means ∆ OAB is an equilateral triangle We need to find area of equilateral triangle Area of equilateral triangle = √𝟑/𝟒 × 𝑺𝒊𝒅𝒆^𝟐 Putting = = √𝟑/𝟒 × 𝑺𝒊𝒅𝒆^𝟐 Area Δ AOB = 1/2 × Base × Height We draw OM ⊥ AB ∴ ∠ OMB = ∠ OMA = 90° And, by symmetry M is the mid-point of AB ∴ BM = AM = 1/2 AB 2 × ∠ OAB + 60° = 180° 2 × ∠ OAB = 180° – 60° 2 × ∠ OAB = 120° ∠ OAB = (120°)/2 ∠ OAB = 60° Thus, ∠ OBA = ∠ OAB = 60° So, all 3 angles are 60° Which means ∆ OAB is an equilateral triangle We need to find area of equilateral triangle Area of equilateral triangle = √𝟑/𝟒 × 𝑺𝒊𝒅𝒆^𝟐 Putting = = √𝟑/𝟒 × 𝑺𝒊𝒅𝒆^𝟐 Area Δ AOB = 1/2 × Base × Height We draw OM ⊥ AB ∴ ∠ OMB = ∠ OMA = 90° And, by symmetry M is the mid-point of AB ∴ BM = AM = 1/2 AB Area of equilateral triangle = √𝟑/𝟒 × 𝑺𝒊𝒅𝒆^𝟐 Putting √3 = 1.73, Side = 15 = (𝟏.𝟕𝟑)/𝟒 × 𝟏𝟓^𝟐 = 1.73/4 × 225 = 1.73/4 × 15^2 = 389.25/4 = 389.25/4 = 97.3125 cm2 Thus, Area of segment APB = Area of sector OAPB – Area of ΔOAB = 117.75 – 97.3125 = 20.4375 cm2 Thus, Area of minor segment = 20.4375 cm2 And, Area of major segment = Area of circle – Area of minor segment = 𝜋𝑟^2 – 20.4375 = 3.14 × 15^2 – 20.4375 = 𝟑.𝟏𝟒 × 𝟐𝟐𝟓 – 20.4375 = 706.5 – 20.4375 = 𝟔𝟖𝟔.𝟎𝟔𝟐𝟓 cm2