To find the area of a triangle in three-dimensional space, we can use the cross product of two vectors that lie in the plane of the triangle. We can use the vectors formed by subtracting the coordinates of one vertex from the other two vertices.
Let's take A as our reference vertex. Then the vectors AB and AC can be calculated as follows:
AB = B - A = (2, 3, 5) - (1, 1, 2) = (1, 2, 3)
AC = C - A = (1, 5, 5) - (1, 1, 2) = (0, 4, 3)
To find the area of the triangle, we need to take the magnitude of the cross product of AB and AC and divide by 2:
Area = |AB x AC| / 2
where x denotes the cross product. The cross product of AB and AC can be calculated as follows:
AB x AC = (23 - 54, 50 - 13, 14 - 20) = (-7, -3, 4)
Taking the magnitude of this vector gives:
|AB x AC| = sqrt((-7)^2 + (-3)^2 + 4^2) = sqrt(74)
Therefore, the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5) is:
Area = |AB x AC| / 2 = sqrt(74) / 2
So, the area of the triangle is (1/2) sqrt(74) square units.