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Example 2 Find the area of the segment AYB shown in figure, if radius of the circle is 21 cm and ∠AOB = 120Β°. (Use Ο€ = 22/7 ). In a given circle, Radius (r) = 21 cm And, 𝜽 = 120Β° Now, Area of segment AYB = Area of sector OAYB – Area of Ξ”OAB Finding Area of sector OAYB Area of sector OAYB = πœƒ/360Γ— πœ‹π‘Ÿ2 = 120/360 Γ— 22/7Γ—(21)2 = 1/3Γ—22/7 Γ— 21 Γ— 21 = 22 Γ— 21 = 462 cm2 Finding area of Ξ” AOB We draw OM βŠ₯ AB ∴ ∠ OMB = ∠ OMA = 90Β° And, by symmetry M is the mid-point of AB ∴ BM = AM = 1/2 AB In right triangle Ξ” OMA sin O = (side opposite to angle O)/Hypotenuse sin πŸ”πŸŽΒ° = 𝐀𝑴/𝑨𝑢 √3/2=𝐴𝑀/21 √3/2 Γ— 21 = AM AM = βˆšπŸ‘/𝟐 Γ— 21 In right triangle Ξ” OMA cos O = (𝑠𝑖𝑑𝑒 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘ π‘‘π‘œ π‘Žπ‘›π‘”π‘™π‘’ 𝑂)/π»π‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ cos πŸ”πŸŽΒ° = 𝑢𝑴/𝑨𝑢 1/2=𝑂𝑀/21 21/2 = OM OM = 𝟐𝟏/𝟐 From (1) AM = 𝟏/𝟐AB 2AM = AB AB = 2AM Putting value of AM AB = 2 Γ— √3/2 Γ— 21 AB = √3 Γ— 21 AB = 21βˆšπŸ‘ cm Now, Area of Ξ” AOB = 1/2 Γ— Base Γ— Height = 𝟏/𝟐 Γ— AB Γ— OM = 1/2 Γ— 21√3 Γ— 21/2 = (πŸ’πŸ’πŸβˆšπŸ‘)/πŸ’ cm2 Therefore, Area of the segment AYB = Area of sector – Area of βˆ† 𝐴𝑂𝐡 = (462 – πŸ’πŸ’πŸ/πŸ’ βˆšπŸ‘ ) cm2

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo