题目

如图，点P是⊙O 外一点，PA切⊙O于点A，AB是⊙O的直径，连接OP，过点B作BC∥OP交⊙O于点C，连接AC交OP于点D． (1)求证：PC是⊙O的切线； (2)若PD=cm，AC=8cm，求图中阴影部分的面积； (3)在(2)的条件下，若点E是的中点，连接CE，求CE的长．     答案：证明： ⑴如图，连接OC， ∵PA切⊙O于A． ∴∠PAO=90º． ····················································································································· 1分 ∵OP∥BC， ∴∠AOP=∠OBC，∠COP=∠OCB． ∵OC=OB， ∴∠OBC=∠OCB， ∴∠AOP=∠COP． ··············································································································· 3分   又∵OA=OC，OP=OP，   ∴△PAO≌△PCO  (SAS)． ∴∠PAO=∠PCO=90 º， 又∵OC是⊙O的半径，   ∴PC是⊙O的切线． ·············································································································· 5分⑵解法一： 由（1）得PA，PC都为圆的切线， ∴PA=PC，OP平分∠APC，∠ADO=∠PAO=90 º， ∴∠PAD+∠DAO=∠DAO+∠AOD， ∴∠PAD =∠AOD， ∴△ADO∽△PDA． ············································································································· 6分 ∴， ∴， ∵AC=8， PD=， ∴AD=AC=4，OD=3，AO=5，····························································································· 7分 由题意知OD为△ABC的中位线， ∴BC=2OD=6，AB=10． ······································································································· 8分 ∴S阴=S半⊙O－S△ACB=． 答：阴影部分的面积为．··················································································· 9分 解法二： ∵AB是⊙O的直径，OP∥BC， ∴∠PDC=∠ACB=90º． ∵∠PCO=90 º， ∴∠PCD+∠ACO=∠ACO+∠OCB=90 º， 即∠PCD=∠OCB． 又∵∠OBC =∠OCB， ∴∠PCD=∠OBC， ∴△PDC∽△ACB， ······································ 6分 ∴． 又∵AC=8， PD=， ∴AD=DC=4，PC=．··················································································· 7分 ∴， ∴CB=6，AB=10， ················································································································ 8分 ∴S阴=S半⊙O-S△ACB=． 答：阴影部分的面积为．··················································································· 9分 （3）如图，连接AE，BE，过点B作BM⊥CE于点M．························································ 10分 ∴∠CMB=∠EMB=∠AEB=90º， 又∵点E是的中点， ∴∠ECB=∠CBM=∠ABE=45º，CM=MB =，BE=ABcos45º=，······························· 11分 ∴ EM=， ∴CE=CM+EM=． 答：CE的长为cm． ····································································································· 12分

查看本题答案和解析