Question

（8分）如圖，在△ABC中，AB=AC，點O為底邊上的中點，以點O為圓心，
1為半徑的半圓與邊AB相切于點D．
（1）判斷直線AC與⊙O的位置關系，并說明理由；
（2）當∠A＝60°時，求圖中陰影部分的面積．

Answer 1

解：（1）直線AC與⊙O相切．························································· 1分
理由是：
連接OD,過點O作OE⊥AC，垂足為點E．

∵⊙O與邊AB相切于點D，
∴OD⊥AB．·························································································· 2分
∵AB=AC，點O為底邊上的中點，
∴AO平分∠BAC····················································································· 3分
又∵OD⊥AB，OE⊥AC
∴OD= OE····························································································· 4分
∴OE是⊙O的半徑．
又∵OE⊥AC，∴直線AC與⊙O相切．·························································· 5分
（2）∵AO平分∠BAC，且∠BAC=60°，∴∠OAD=∠OAE=30°，
∴∠AOD=∠AOE=60°，

解析