The area of a triangle can be calculated using the formula:
$$A = \frac{1}{2} \cdot b \cdot h$$
where $b$ is the base of the triangle and $h$ is the height of the triangle.
In this case, the base of the triangle is the line segment connecting $(1, 0)$ and $(2, 2)$. The length of this line segment can be calculated using the Pythagorean theorem:
$$|(x_1 – x_2)|^2 + |(y_1 – y_2)|^2 = d^2$$
where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two endpoints of the line segment, and $d$ is the length of the line segment.
In this case, we have:
$$|(1 – 2)|^2 + |(0 – 2)|^2 = d^2$$
$$1^2 + 4^2 = d^2$$
$$d = \sqrt{1 + 16} = \sqrt{17}$$
The height of the triangle is the perpendicular distance from $(1, 0)$ to the line segment connecting $(2, 2)$ and $(4, 3)$. This can be calculated using the formula:
$$h = \sqrt{(x_1 – x_2)^2 + (y_1 – y_2)^2}$$
In this case, we have:
$$h = \sqrt{(1 – 2)^2 + (0 – 2)^2} = \sqrt{5}$$
Therefore, the area of the triangle is:
$$A = \frac{1}{2} \cdot \sqrt{17} \cdot \sqrt{5} = \frac{3}{2}$$
Therefore, the correct answer is $\boxed{\frac{3}{2}}$.