In the figure, O is the centre of semicircle. Triangle ABC is right at B. Side AB and side BC touch the semicircle at points D and E respectively . If OA = 15 cm , OC = 20 cm then find the radius of the semicircle .
Answers
Radius of Semicircle = 12 cm
Step-by-step explanation:
OD = OE = r = Radius
as DB & DE are tangent
Hence right angle
∠B = 90°
=> ∠BOE = 90°
DOEB is square
DB = BE = r
AD² = AO² - OD² = 15² - r² = 225 - r²
CE² = CO² - OE² = 20² - r² = 400 - r²
AB = AD + DB = √(225 - r²) + r
BC = CE + BE = √(400 - r²) + r
AB² + BC² = AC²
=> (√(225 - r²) + r)² + (√(400 - r²) + r)² = (15 + 20)²
=> 225 - r² + r² + 2r√(225 - r²) + 400 - r² + r² + 2r√(400 - r²) = 1225
=> 2r√(225 - r²) + 2r√(400 - r²) = 600
=> r√(225 - r²) + r√(400 - r²) = 300
Solving r = 12
Radius of Semicircle = 12 cm
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