题目

如图，AB=16，O为AB中点，点C在线段OB上（不与点O，B重合），将OC绕点O逆时针旋转270°后得到扇形COD，AP，BQ分别切优弧于点P，Q，且点P，Q在AB异侧，连接OP． （1）求证：AP=BQ； （2）当BQ=4时，求扇形COQ的面积及的长（结果保留π）； （3）若△APO的外心在扇形COD的内部，请直接写出OC的取值范围． 答案：（1）证明：连接OQ，如图所示． ∵AP、BQ是⊙O的切线， ∴OP⊥AP，OQ⊥BQ， ∴∠APO=∠BQO=90°． 在Rt△APO和Rt△BQO中，， ∴Rt△APO≌Rt△BQO（HL）， ∴AP=BQ． （2）解：∵Rt△APO≌Rt△BQO， ∴∠AOP=∠BOQ， ∴P、O、Q三点共线． ∵在Rt△BOQ中，cosB===， ∴∠B=30°，∠BOQ=60°， ∴OQ=OB=4， ∴S扇形COQ==π． ∵∠COD=90°，阅读理解。     My friend has a large dog named Jack. Every afternoon my ffiend takes it for a walk. Jack likes the walk in the park very much.     One Sunday afternoon a man came to visit my friend. He stayed a long time. He talked and talked. Soon it was time for my friend to take Jack for his walk. Jack worried about this. It walked around the room several times and then sat down in front of the visitor and looked at him, but he kept on talking. Jack couldn't stand it any longer. He went out of the room and came back a moment later. He sat down again in front of the visitor, holding the visitor's hat in his mouth. Seeing this, the visitor couldn't help laughing. He stood up and said good-bye to my friend in a hurry.根据短文内容，完成下列句子。1. The dog, Jack,liked            very much in the park.2. Every             my friend takes the dog for a walk.3. Jack sat down again in front of             holding the visitor's hat in its mouth and looked at him in order     to tell the visitor that it was time for him to say good-bye.4.The second paragraph tells us that             is very clever.