Question

【题目】（本题满分10分）

（1）如图①，在正方形ABCD中，△AEF的顶点E，F分别在BC，CD边上，高AG与正方形的边长相等，求的度数．

（2）如图②，在Rt△ABD中，，，点M，N是BD边上的任意两点，且，将△ABM绕点A逆时针旋转至△ADH位置，连接，试判断MN，ND，DH之间的数量关系，并说明理由．

（3）在图①中，连接BD分别交AE，AF于点M，N，若，，，求AG，MN的长．

Answer 1

【答案】（1）在Rt△ABE和Rt△AGE中，，，

∴△ABE≌△AGE． ∴．············································1分

同理，．

∴．·····························································2分

（2）．··························································3分

∵，，

∴． ∴．

又∵，，

∴△AMN≌△AHN． ∴．············································5分

∵，，

∴． ∴．

∴． ∴．·····················································6分

（3）由（1）知，，．

设，则，．

∵，

∴．

解这个方程，得，（舍去负根）．

∴．·····························································8分

∴．

在（2）中，，，

∴．·····························································9分

设，则．

∴．即．·························································10分

【解析】略