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A village has a circular wall around it, and the wall has four gates pointing north, south, east and west. A tree stands outside the village. 16 m north of the north gate, and it can be justseen appearing on the horizon from a point 48 m east of the south gate. What is the diameter,in meters, of the wall that surrounds the village? |
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Answer» Diameter of Wall = 48 mExplanation:A village has a circular wall Let say Diameter of Wall = 2RThen Radius = RLet say Tree is at G = 16 m North Of North Gate NG = 16 m& point of Observation = P 48m east of the south gate PS = 48mPS is Tangentnow Tree appearing on the horizon so Let say PQ WOULD be Tangent which on extend goes upto GPQ = PS = 48 ( equal Tangent)Let say O is CENTER of wallthen OG² = QG² + OQ²OQ = Radius = Rlet say QG = XOG = R + 16(R + 16)² = X² + R² => X² = 32R + 256=> X² = 32(R + 8) - eq 1in ΔPSGPG² = PS² + SG²=> (48 + X)² = 48² + (2R + 16)²=> 48² + X² + 96X = 48² + 4R² + 256 + 64R=> 32R + 256 + 96X = 4R² + 256 + 64R=> 96X = 4R² + 32R=> 24X = R² + 8R=> 24X = R(R + 8) Squaring both Sides=> 24²X² = R²(R + 8)²R²(R + 8)² = 24² * 32(R + 8)=> R² (R + 8) = 24² * (24 + 8)=> R = 24Diameter of Wall = 2R = 2 * 24 = 48 mDiameter of Wall = 48 m |
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