Question

如圖1，矩形MNPQ中，點E，F，G，H分別在NP，PQ，QM，MN上，若，則稱四邊形EFGH為矩形MNPQ的反射四邊形．圖2，圖3，圖4中，四邊形ABCD為矩形，且，．

理解與作圖：

（1）在圖2、圖3中，點E，F分別在BC，CD邊上，試利用正方形網格在圖上作出矩形ABCD的反射四邊形EFGH．

計算與猜想：

（2）求圖2，圖3中反射四邊形EFGH的周長，并猜想矩形ABCD的反射四邊形的周長是否為定值？

啟發與證明：

（3）如圖4，為了證明上述猜想，小華同學嘗試延長GF交BC的延長線于M，試利用小華同學給我們的啟發證明（2）中的猜想．

Answer 1

1）作圖如下：······································································································· 2分

（2）解：在圖2中，，

∴四邊形EFGH的周長為．············································································ 3分

在圖3中，，．∴四邊形EFGH的周長為．······················································································ 4分

猜想：矩形ABCD的反射四邊形的周長為定值．···················································· 5分

（3）如圖4，證法一：延長GH交CB的延長線于點N．

∵，，

∴．

而，

∴Rt△FCE≌Rt△FCM．

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

同理：，．

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

∵，，

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

過點G作GK⊥BC于K，則．······················································ 9分

∴．

∴四邊形EFGH的周長為．······························································ 10分

證法二：∵，，    ∴．

而，    ∴Rt△FCE≌Rt△FCM．

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

∵，，

而，   ∴．

∴HE∥GF．    同理：GH∥EF．

∴四邊形EFGH是平行四邊形．

∴．     而，

∴Rt△FDG≌Rt△HBE．     ∴．

過點G作GK⊥BC于K，則

∴．

∴四邊形EFGH的周長為．