✔ 最佳答案
1.
(a)
∠OAD
= (1/2) x (∠CAB - ∠DAB)
= (1/2) x (78° - 34°)
= 22°
在 ΔOAD 中:
sin∠OAD = OD/OA
sin22° = r/OA
OA = r/sin22°
∠OBE
= (1/2) x (∠FBA - ∠EBA)= (1/2) x (102° - 50°)
= 26°
在 ΔOBE 中:
sin∠OBE= OE/OB
sin26° = r/OB
OB = r/sin26°
(b)
∠OAB
= ∠OAD+ ∠DAB
= 22° + 34°
= 56°
∠OBA
= ∠OBE+ ∠EBA
= 26° + 50°
= 76°
在 ΔAOB 中:
∠AOB+ ∠OAB+ ∠OBA= 180°
∠AOB+ 56° + 76° = 180°
∠AOB= 48°
在 ΔAOB 中:
AB² = OA² + OB² - 2 x OA x OB x cos∠AOB
120² = (r/sin22°)² + (r/sin26°)² - 2 x (r/sin22°) x (r/sin26°) x cos48°
r = 59
半徑 = 59 米
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2.
由角平分線的性質:
AB : AC = BD : DC
BD : DC = 3 : 4
設 AD = d
設 AB = 3k,則 DC = 4k
在 ΔABD 中:
BD² = AB² + AD² - 2 x AB x AD x cos60° (餘弦定律)
(3k)² = 3² + d² - 2 x 3 x d x (1/2)
9k² = d² - 3d + 9
144k² = 16d² - 48d + 144 …… [1]
在 ΔACD 中:
DC² = AC² + AD² - 2 x AC x AD x cos60° (餘弦定律)
(4k)² = 4² + d² - 2 x 4 x d x (1/2)
16k² = d² - 4d + 16
144k² = 9d² - 36d + 144 …… [2]
[1] - [2] :
7d² - 12d = 0
d(7d - 12) = 0
d = 0 (不合) 或 d = 12/7
所以 AD = 12/7
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3.
聯 AC、OA及 OC。
在圓內接四邊形中,對角和 ∠B + ∠D = 180°
120° + ∠D = 180°
∠D =60°
三角形內角和為 180°,在等腰 ΔOAC 中:
∠BAC= ∠BAC= (180° - 120°)/2 = 30°
∠DAC+ ∠BAC=∠BAD
∠DAC+ 30° =75°
∠DAC= 45°
ΔACD 內角和為 180°:
∠D +∠ACD+ ∠CAD= 180°
60° + ∠ACD + 45° = 180°
∠ACD= 75°
圓心角 ∠AOC = 2∠D = 2 x 60° = 120°
在 ΔOAC 中:
OC / sin∠OAC = AC / sin∠AOC
r / sin30° = AC / sin120°
AC = r sin120° / sin30° = (√3)r
在 ΔACD 中:
sin∠D /AC = sin∠CAD / CD
sin60° / (√3)r = sin45° / CD
CD = (√3)r x sin45° / sin60°
CD = (√2)r = 1.414r (四位有效數字)
在 ΔACD 中:
sin∠D /AC = sin∠ACD / CD
sin60° / (√3)r = sin75° / AD
CD = (√3)r x sin75° / sin60°
CD = [(√2)+ (√6)]r/2 = 1.932r (四位有效數字)
∠B =∠AOC= 120°
∠OCA= ∠BCA= 30°
∠OAC= ∠BAC= 30°
ΔOCA ≡ ΔBCA (ASA)
全等三角形對應邊相等:
BC = OC = r
AB = AO = r