For ease of reference I will label the vertices:
#color(white)("XXX")A: (4,1)#
#color(white)("XXX")B: (3,4)#
#color(white)("XXX")C: (-3,2)#
#color(white)("XXX")D: (-2,-1)#
Notice that the slopes of #AB# and #CD# are both #-3#
#color(white)("XXX")#You can determine this, for example, for #AB# by
#color(white)("XXX")m_(AB)=(Deltay_(AB))/(Deltax_(AB))=(4-1)/(3-4)=-3#
Similarly we can note that the slopes of #BC# and #DA# are both #1/3#.
Since the slopes of #AB# and #CD# are negative reciprocals of #BC# and #DA#,
#AB# and #CD# are perpendicular to #BC# and #DA#.
#rArr ABCD# is a rectangle.
The area of #ABCD# can be calculated as the length #abs(AB)# ties the length #abs(CD)#
Using the Pythagorean Theorem:
#abs(AB)=sqrt(Deltax_(AB)^2+Deltay_(AB)^2)#
#color(white)(abs(AB))=sqrt(1^1+3^2)=sqrt(10)#
and
#abs(CD)=sqrt(Delta_(CD)x^2+Deltay_(CD)^2)#
#color(white)(abs(CD))=sqrt(2^2_6^2)=sqrt(40)=2sqrt(10)#
Therefore
#"Area"_(ABCD)=sqrt(10)xx2sqrt(10)=20#
