Question

在四边形ABCD中,∠BAD=∠ACD=90°,AB=AD,AC=4BC,

设CD的长为X，四边形ABCD的面积为Y、则Y与X之间的函数关系
图片

Answer 1

如图，设BC=a ,则 AC=4a, 过点D作AC的垂线交AC于K

                       则，∠AKD=90°，即∠2+∠3=90°

                           又∵∠BAD=90°，即∠1+∠3=90°

                               ∴ ∠1=∠2

                          又∵  ∠BCA=∠AKD=90°

                                      AB=AD

                               ∴Rt△BCA≌△AKD （角、角、边）

                ∴ AK=BC=a     DK=AC=4a ,

                               ∵ KC=AC-AK=4a-a=3a

                            在Rt△DKC中，DC²=KC²+DK²=9a²+16a²=25a²

                         ∴ DC=5a

                                           ∵DC=X

                                          ∴ 5a=X ,即 a=X/5

                                S四边形ABCD=S△ABC+S△ACD=BC*AC/2+AC*DK/2=a*4a/2+4a*4a/2=10a²

                      ∴Y=10(X/5)²=2X²/5

                                  ∴ Y与X之间的关系是：Y=2X²/5