For ease of reference I will label the vertices:
color(white)("XXX")A: (4,1)XXXA:(4,1)
color(white)("XXX")B: (3,4)XXXB:(3,4)
color(white)("XXX")C: (-3,2)XXXC:(−3,2)
color(white)("XXX")D: (-2,-1)XXXD:(−2,−1)
Notice that the slopes of ABAB and CDCD are both -3−3
color(white)("XXX")XXXYou can determine this, for example, for ABAB 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
