F.4 Amaths question (trigonometry)

wai shing

2006-12-30 8:38 pm

題目是這樣的︰
In the figue, BA and BC are tangents to the circle with centre at O. ∠ABC = 2┬┬ / 3. If the perimemter of quadrilateral OABC is longer than the preimemter of sector OAC by 1 cm, find the radius of the circle.
[hint: OC ┴ CB, OA ┴AB]

我畫了那幅圖︰http://hk.geocities.com/frederickshing/reex5no2.gif
(第一次可能會load唔到，load多幾次就得架啦﹗)

答案是︰
6(√3)+ 3┬┬
----------------- cm
12 - ┬┬2

…毫無頭緒...
P.S.( ┬┬ = pie)

請各位好心人教教我吧﹗最好附上少少解釋，多謝﹗^^

回答 (1)

Marie

2006-12-30 9:49 pm

✔ 最佳答案

∠AOC = 2π - π - 2π/3
∠AOC = π/3

preimemter of sector OAC
= 2πr [(π/3)/2π] + 2r
= πr/3 + 2r

∠OBC = ∠OBA = (2π/3) / 2 = π/3
CB = BA = r tan(π/3)
perimemter of quadrilateral OABC = 2r tan (π/3) + 2r

2r tan(π/3) + 2r - πr/3 - 2r = 1
r [2 tan(π/3) - π/3] = 1
r = 1 / [2√3 - π/3]
r = (2√3 + π/3) / 12 - (π^2/9)
r = (18√3 + 3π) / (108 - π^2)

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