數學題求解

1)√65 - √63
= (√65 - √63) (√65 + √63) / (√65 + √63)
= (65 - 63) / (√65 + √63)
= 2 / (√65 + √63)
= 2 / √(65 + 63 + 2√(65×63))
= 2 / √(128 + 2√(64² - 1))
> 2 / √(128 + 2√64²) = 1/8 = 0.125 , 故較接近 0.13 , 選(B)。
2)延長 BA、CD 交於 E , 則 ∠ EAD = 45° , ∠ EDA = 75° , ∠AED = 60°。
由正弦定理 , ED : sin45° = AD : sin60° = EA : sin75°
ED : √2/2 = 3√2 : √3/2 = EA : sin(45°+30°)
ED : √2/2 = 2√6 = EA : (sin45°cos30° + cos45° sin30°)
ED : √2/2 = 2√6 = EA : (√2/2 × √3/2 + √2/2 × 1/2)
ED : √2/2 = 2√6 = EA : (√6 + √2)/4
∴ ED = 2√3 , EA = 3 + √3R
= △EBC - △EAD
= (1/2)sin60° EB EC - (1/2)sin60° EA ED
= (√3/4) ( (9 + √3) 8√3 - (3 + √3) 2√3 )
= (3/2) ( (9 + √3)4 - (3 + √3) )
= (9/2) (11 + √3)
= 57.294... , 選(A)。


2013-12-19 16:33:36 補充：
∠ EAD = 180° - ∠ BAD = 180° - 135° = 45°
∠ EDA = 180° - ∠ ADC = 180° - 105° = 75°