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:略