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