中五級數學-圓

1.
∠QPS + ∠QRS = 180° (圓內接四邊形內對角)
101° + ∠QRS = 180°
∠QRS = 79°

∠POR = 2∠RST (圓心角為對同弧圓心角兩倍)

∠POR + ∠STR = 180° (圓內接四邊形內對角)
2∠RST + ∠STR = 180°
∠STR = 180° - 2∠RST

∠RST + ∠STR + ∠QRS = 180° (Δ內角和)
∠RST + (180° - 2∠RST)+ 79° = 180°
∠RST = 79°

∠STR = 180° - 2∠RST
∠STR = 180° - 2 × 79°
∠STR = 22°


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2.
(a)
tan∠MOB = MB/OM
tan75° = MB/OM
MB = OM × tan75°

ΔMOB面積：
(1/2) × OM × (OM × tan75°) = 50 cm²
OM² tan75° = 100 cm²
y = √(100/tan75°) cm

cos∠MOB = OM/OB
cos75° = OM/OB
OB = [√(100/tan75°)] / cos75° cm
OB = 20 cm

OE = OB - BE
OE = 20 - 3 cm
小圓半徑 = 17 cm

(b)
連OD。

在ΔOMD中：
MD² = OD² - OM² (畢氏定理)
MD² = 17² - (100/tan75°)cm²
MD = √[17² - (100/tan75°)] cm

CD = 2 MD
CD = 2 × √[17² - (100/tan75°)] cm
CD = 32.4 cm (至三位有效數字)


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3.
(a)
對孤CD的圓心角
= 20° × 2
= 40°

弧CD長度：
圓周 × (40/360) = 4 cm
圓周 = 36 cm

弧AB
= (1/2) × 36 - 4 - 11 cm
= 3 cm

(b)(i)
圓直徑AD
= 36/π cm
= 11.46 cm (至二位小數)

(b)(ii)
設 O 為圓心。

弧AB所對之圓心角 ∠AOB
= 360° × (3/36)
= 30°

圓半徑
= (36/π) ÷ 2
= 18/π

在ΔAOB中：
sin(∠AOB/2) = (1/2)AB / (18/π)
sin15° = (1/2)AB / (18/π cm)
AB = 2 × (18/π) × sin15° cm
AB = 2.97 cm (至二位小數)