Let
\(A(0,0)\\ B (2.8, 2.1)\\C(4.3, 4.4)\)
using a graph tool
see the attached figure
The figure is a triangle
we know that
The Heron's Formula is a method for calculating the area of a triangle when you know the lengths of all three sides.
Hero's Formula is equal to
\(Area=\sqrt{p*(p-a)*(p-b)*(p-c)}\)
where
p is is half the perimeter of the triangle
a,b,c are the lengths of the sides of a triangle
so
Step \(1\)
Find the length sides of the triangle
a) Find the distance AB
\(d =\sqrt{(y2-y1)^{2} +(x2-x1)^{2}}\)
Substitute
\(d =\sqrt{(2.1-0)^{2} +(2.8-0)^{2}}\)
\(d =3.5\ mi}\)
b) Find the distance AC
\(d =\sqrt{(y2-y1)^{2} +(x2-x1)^{2}}\)
Substitute
\(d =\sqrt{(4.4-0)^{2} +(4.3-0)^{2}}\)
\(d =6.15\ mi}\)
c) Find the distance BC
\(d =\sqrt{(y2-y1)^{2} +(x2-x1)^{2}}\)
Substitute
\(d =\sqrt{(4.4-2.1)^{2} +(4.3-2.8)^{2}}\)
\(d =2.75\ mi}\)
Step \(2\)
Find the perimeter of the triangle
\(Perimeter= 3.5+6.15+2.75 =12.4\ mi\)
Find the half of the perimeter
\(p=\frac{12.4}{2} = 6.2\ mi\)
Step \(3\)
Find the area of the triangle
\(Area=\sqrt{6.2*(6.2-3.5)*(6.2-6.15)*(6.2-2.75)}\)
\(Area=\sqrt{6.2*(2.7)*(0.05)*(3.45)}\)
\(Area= 1.69\ mi^{2}\)
therefore
the answer is the option
a) 1.65 mi^2.
