Three consecutive vertices of a parallelogram are (– 2, 1); (1, 0) and (4, 3). Find the coordinate of the fourth vertex and find the area of the parallelogram.

D(1,4) - the fourth vertex, the area is equal to 12

Step-by-step explanation:

A(2,-1), B(1,0),C(4,3), They are consecutive, we need to find the point D, find the midpoint of AC (O)

xo= (xA+xc)/2    x0= (-2+4)/2=1

y0=(yA+yC)/2    y0= (1+3)/2=2

0(1,2)

O is the midpoint of BD, so let's find the vertex D

X0= (xB+xD)/2     1= (1+xD)/2   xD=1

y0= (yB+yD)/2      2= (0+yD)/2    yD=4

D(1,4)

the vector DA is (-2-1, 1-4)= (-3,-3)

the vector DC is (4-1, 3-4)= (3,-1)

The modul of DC is sqrt ((-3)^2+(-3)^2)= 3*sqrt2(the length of the side Dc)

The modul of DA is sqrt (3^2+(-1)^2)= sqrt10(the length of the side DA)

cosD= (-3*3+(-3)(-1))/3*sqrt2*sqrt10=-6/3*2*sqrt5=-1/sqrt5

sinD= sqrt (1- 1/5)=2/ sqrt5

S=3sqrt2*sqrt10*2/sqrt5= 12