Since the problem provides the vertices of triangle, you may use the formula that involves determinants such that:
Hence, evaluating the area of the triangle yields A = 44.
We can calculate the area of the triangle, using Heron's formula.
In Heron's formula, we have to know the lengths of the sides of the triangle. Since we know the coordinates of the vertices of the triangle, we can calculate the lengths of the sides.
We'll use the following formula to determine the lenght of the side:
AB = sqrt[(xB-xA)^2 + (yB-yA)^2]
We'll calculate the length of the side, AB, whose vertices are the points A(0,5) and B( 8,2).
(0,16) , (0,5) and (8,2)
AB = sqrt[(8-0)^2 + (2-5)^2]
AB = sqrt(64 + 9)
AB = sqrt 73
We'll calculate the length of the side, AC, whose vertices are the points A(0,5) and C(0,16).
AC = sqrt[(0-0)^2 + (16-5)^2]
AC = 11
We'll calculate the length of the side, BC, whose vertices are the points B( 8,2) and C(0,16).
BC = sqrt[(0-8)^2 + (16-2)^2]
BC =sqrt(64 + 196)
BC = sqrt260
We'll apply Heron's formula:
S = sqrt[p(p-AB)(p-AC)(p-BC)]
p = (AB+AC+BC)/2
p = (sqrt73+11+sqrt260)/2