一條台灣中三考卷中出現的考題。

2007-12-08 5:08 am

<img src=\"http://s120.photobucket.com/albums/o179/bloodavril_2006/?action=view&current=2108134216.jpg\">

這是一條我在網上看到的數學題目，
找不到答案，知道須用未知數在圖上標示，
請各位幫幫我，在圖上標示未知數的位置，或清楚表達！

感謝萬分！

回答 (1)

用戶82723

2007-12-08 9:04 pm

✔ 最佳答案

圖片參考：http://i207.photobucket.com/albums/bb173/kkkpopup/area1.gif

Let y be the length of side of square ABCD.

Since AD = AN = ND (radii),
∠NAD = 60°
Similarly, ∠BAE = 60°
Hence, ∠NAE = 30°
By Cosine Law,
NE = √ [y2 + y2 - 2(y)(y)cos 30° ]
= √(2 - √3) y
Area of segment NE
= Area of sector ANE - Area of triangle ANE
= (y2π)(30°/360°) - (y2 sin 30°)/2
= (π/12 - 1/4) y2
By symmetry, EF = FM = MN = NE
and area of segment EF = area of segment FM = area of segment MN = area of segment NE
Therefore, area of the shaded region
= [√(2 - √3) y]2 + 4(π/12 - 1/4) y2
= (2 - √3 + π/3 - 1)y2
= (1 - √3 + π/3)y2

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