Question

（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

（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