amaths

(a)
The mid-point of AC
=((3+7)/2,(-1+11)/2)=(5,5)
The mid-point of BD
=((15-5)/2,(7+3)/2)=(5,5)
Since the diagonals AC and BD bisect each other (intersection point E(5,5)) and so ABCD is a parallelogram
(b)
AC=(7-3,11+1)=(4,12)
DB=(15-(-5),7-3)=(20,10)
|AC|=√160
|DB|=√500
AC.DB=4*20+12*10=200
Let acute angle between the diagonals AC and BD is x
cosx=AC.DB/|AC||DB|
cosx=200/√80000=1/√2
x=45
The acute angle between the diagonals AC and BD is 45 degrees


2007-11-23 19:38:58 補充：
用向量DB代替BD是為了方便運算