如图:正方形ABCD中,过点D作DP交AC于点M、交AB于点N,交CB的延长线于点P,若MN=1,PN=3,则DM的长为_.

问题描述:

如图:正方形ABCD中,过点D作DP交AC于点M、交AB于点N,交CB的延长线于点P,若MN=1,PN=3,则DM的长为______.

设DM=x,PB=y,正方形ABCD的边长是Z,则DN=x+1
∵AD∥PC
∴△ADM∽△CPM,△PNB∽△DNA

PN
DN
=
PB
AD

3
x+1
=
y
z
4
x
=
y+z
z
=
y
z
+1,
3
x+1
=
4
x
-1,
3x=4(x+1)-x2-x,
x=2或-2(舍去),
∴x=2.