F.4 trigon isosceles triangle

[匿名]

2013-06-13 12:17 am

Figure 2 shows an isosceles triangle with AC=BC=x and AB=6.
If D is a point on AB such that angleACD= 0 and angleBCD= 20.
a)show that AD =6/(1+2cos0).
b) find the range of AD.
http://upload.lsforum.net/users/public/x15601mathg66.jpg

回答 (1)

用戶2053

2013-06-13 1:35 am

✔ 最佳答案

圖片參考：http://upload.lsforum.net/users/public/x15601mathg66.jpg

a)

By sine formula ,
AD / sinθ = x / sin∠CDA ... (1)
and
BD / sin2θ = x / sin∠CDB
(6 - AD) / sin2θ = x / sin(180° - ∠CDA)
(6 - AD) / (2 sinθ cosθ) = x / sin∠CDA
(6 - AD) / (2 sinθ cosθ) = AD / sinθ ... by (1)
6 - AD = 2 cosθ AD
AD = 6 / (1 + 2cosθ)

b)

0 < 3θ = ∠ACB < 180°
0 < θ < 60°
∴
6 / (1 + 2cos0°) < 6 / (1 + 2cosθ) < 6 / (1 + 2cos60°)
6 / (1 + 2) < AD < 6 / (1 + 2(1/2))
2 < AD < 3

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