Geo-gebra 2
on the given figure, a circle w with centre at O and diameter AB = 2R. inside the circle there are two similar circular arcs of radius R with centres at A,B and they intersect at O.if the green area is H and the area of the circle w is F, then H/F = ( √a – π) / 3π. Find the value of a.
Notation: The two arcs and the diameter AB all intersect at O.
The answer is 27.
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Let the circles centres at O,A,B ,(the radius of All circles is R ) be : (0,0 ) , (0,R ) , (0,-R ) respectively. hence their equations will be : x² + y²=R² , x² + (y-R)² = R² , x² + (y+R)² = R² respectively. Now we can use these equations to determine the points of intersections N,P,M,and Q. From these coordinates we compute the following lengths : NP = MQ = √3R , PQ = NM = R. NMPQ Area = √3R² (rectangle). Area of sectors NOP + MOQ = A1 = 2R²(π/3-√3/4) , Similarly Area of sectors made by NM and PQ = A2 = 2R²(π/6-√3/4) , The green aera H = MNPQ – A1 + A2 = R²(√3-π/3). F = πR² , H/F = (3√3-π)/3π = (√27-π)/3π.