Question

（本題滿分10分）
（1）如圖①，在正方形ABCD中，△AEF的頂點E，F分別在BC，CD邊上，高AG與正方形的邊長相等，求的度數．
（2）如圖②，在Rt△ABD中，，，點M，N是BD邊上的任意兩點，且，將△ABM繞點A逆時針旋轉至△ADH位置，連接，試判斷MN，ND，DH之間的數量關系，并說明理由．
（3）在圖①中，連接BD分別交AE，AF于點M，N，若，，，求AG，MN的長．
        

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，，．
設，則，．
∵，
∴．
解這個方程，得，（舍去負根）．
∴．························································································· 8分
∴．
在（2）中，，，
∴．·········································································· 9分
設，則．
∴．即．···································································· 10分

解析