中二數學一題

OA:AC = 2:1

詳盡的ans.

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2009-04-14 13:42:19 補充：
簡單又易明

2009-04-14 13:43:12 補充：
simple ans.

Let OA = x, OC = y and angle AOB = angle COD = a degree.

Then, the area of the sector AOB = (a / 2π)*πx^2
_________________________= (ax^2) / 2

The area of the sector COD = (a / 2π)*πy^2
_____________________= (ay^2) / 2

The area of the shaded region = [(ay^2) / 2] - [(ax^2) / 2]
_______________________= (a/2)(y^2 - x^2)

(a/2)(y^2 - x^2) : (ax^2) / 2 = 5 : 4
_______(y^2 - x^2) / x^2 = 5 / 4
___________4y^2 - 4x^2 = 5x^2
________________4y^2 = 9x^2
_____________(y / x)^2 = 9 / 4
_________________y / x = 3 / 2 or -3 / 2 (rejected as x & y are +ve.)
_________________y : x = 3 : 2

So, OC : OA = 3 : 2

Hence, OC = 3k, OA = 2k & AC = OC - OA = 3k - 2k = k

Therefore, OA : AC = 2k : k = 2 : 1