The area of a triangle given its vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ can be calculated using the formula:

$\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$

Here, the vertices of the triangle are $A(12, 4)$, $B(1, 1)$, and $C(0, 8)$. Plugging the coordinates into the formula:

$\text{Area} = \frac{1}{2} \left| 12(1 - 8) + 1(8 - 4) + 0(4 - 1) \right|$

Simplifying step-by-step:

$\text{Area} = \frac{1}{2} \left| 12(-7) + 1(4) + 0 \right|$ $\text{Area} = \frac{1}{2} \left| -84 + 4 \right|$ $\text{Area} = \frac{1}{2} \left| -80 \right|$ $\text{Area} = \frac{1}{2} \times 80 = 40$

Thus, the area of the triangle is $40$ square units.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How would the formula change if the triangle vertices were in 3D?
2. Can the area be negative? Why or why not?
3. How can you verify the triangle's area using Heron's formula?
4. What happens if two points of the triangle overlap?
5. How do you find the centroid of this triangle using its vertices?

Tip: Always make sure to calculate absolute values when applying the triangle area formula to avoid negative areas!