Question

【題目】（本題滿分10分）

（1）如圖①，在正方形ABCD中，△AEF的頂點(diǎn)E，F分別在BC，CD邊上，高AG與正方形的邊長(zhǎng)相等，求的度數(shù)．

（2）如圖②，在Rt△ABD中，，，點(diǎn)M，N是BD邊上的任意兩點(diǎn)，且，將△ABM繞點(diǎn)A逆時(shí)針旋轉(zhuǎn)至△ADH位置，連接，試判斷MN，ND，DH之間的數(shù)量關(guān)系，并說(shuō)明理由．

（3）在圖①中，連接BD分別交AE，AF于點(diǎn)M，N，若，，，求AG，MN的長(zhǎng)．

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，，．

設(shè)，則，．

∵，

∴．

解這個(gè)方程，得，（舍去負(fù)根）．

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

∴．

在（2）中，，，

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

設(shè)，則．

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

【解析】略