已知圓內接四邊形 各邊長 ab線段等於3 bc線段等於3 cd線段等於5 da線段等於8 請問此圓的半徑等於多少?

圓內接四邊形, AB=BC=3, CD=5, DA=8, 半徑r=?


Cosine Law:

AC^2 = 9 + 9 - 18cosB = 25 + 64 - 80cosD

18 - 18cosB = 89 - cos(180 - B) = 89 + 80cosB

=> 89cosB = -71

=> cosB = -71/89

=> B = 136.4264°


O = Center

OH⊥BC


ΔOBC: BH = HC = 1.5, OA = OB = OC = r

∠OBC = ∠OBA = 136.426/2 = 68.2132°

=> r = 1.5/cos68.2132° = 4.04145.....Answer