Answer:
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
