题目内容
(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.
(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