Question

如图1，矩形MNPQ中，点E，F，G，H分别在NP，PQ，QM，MN上，若，则称四边形EFGH为矩形MNPQ的反射四边形．图2，图3，图4中，四边形ABCD为矩形，且，．

理解与作图：

（1）在图2、图3中，点E，F分别在BC，CD边上，试利用正方形网格在图上作出矩形ABCD的反射四边形EFGH．

计算与猜想：

（2）求图2，图3中反射四边形EFGH的周长，并猜想矩形ABCD的反射四边形的周长是否为定值？

启发与证明：

（3）如图4，为了证明上述猜想，小华同学尝试延长GF交BC的延长线于M，试利用小华同学给我们的启发证明（2）中的猜想．

Answer 1

1）作图如下：······································································································· 2分

（2）解：在图2中，，

∴四边形EFGH的周长为．············································································ 3分

在图3中，，．∴四边形EFGH的周长为．······················································································ 4分

猜想：矩形ABCD的反射四边形的周长为定值．···················································· 5分

（3）如图4，证法一：延长GH交CB的延长线于点N．

∵，，

∴．

而，

∴Rt△FCE≌Rt△FCM．

∴，．··················································································· 6分

同理：，．

∴．······························································································ 7分

∵，，

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

过点G作GK⊥BC于K，则．······················································ 9分

∴．

∴四边形EFGH的周长为．······························································ 10分

证法二：∵，，    ∴．

而，    ∴Rt△FCE≌Rt△FCM．

∴，．··················································································· 6分

∵，，

而，   ∴．

∴HE∥GF．    同理：GH∥EF．

∴四边形EFGH是平行四边形．

∴．     而，

∴Rt△FDG≌Rt△HBE．     ∴．

过点G作GK⊥BC于K，则

∴．

∴四边形EFGH的周长为．