To find the area of a triangle given its vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$, we can use the formula:

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

Substitute the given coordinates of the vertices:

• $A(12,4)$
• $B(1,1)$
• $C(0,8)$

The formula becomes:

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

Now, calculate each part:

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

Thus, the area of the triangle is $\boxed{40}$ square units.

Would you like further clarification or more details on this? Here are some related questions:

1. How do you find the area of a triangle using the distance formula between points?
2. How does the shoelace (or determinant) formula relate to the area of polygons?
3. Can the area of a triangle be negative? If so, what does it represent?
4. How can this method be adapted to find the area of a quadrilateral?
5. What happens if two of the points are collinear?

Tip: When working with triangles, ensure the vertices are not collinear by checking the area. If it's zero, the points are on the same line!