To find the area of a triangle given the coordinates of its vertices, we can use the formula:

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

Where $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ are the coordinates of the vertices.

Given vertices:

• $A(12, 4)$
• $B(1, 1)$
• $C(0, 8)$

Let’s plug the coordinates into the formula.

$\text{Area} = \frac{1}{2} \left| 12(1 - 8) + 1(8 - 4) + 0(4 - 1) \right|$ $= \frac{1}{2} \left| 12(-7) + 1(4) + 0(3) \right|$ $= \frac{1}{2} \left| -84 + 4 + 0 \right|$ $= \frac{1}{2} \left| -80 \right|$ $= \frac{1}{2} \times 80 = 40$

The area of the triangle is $40$ square units.

Would you like more details or have any other questions?

Here are some related questions:

1. How can you find the area of a triangle using side lengths?
2. What is the determinant method for calculating the area of a triangle?
3. How do you calculate the area of a triangle using Heron’s formula?
4. Can you use vectors to find the area of a triangle?
5. How do transformations affect the area of a triangle?

Tip: For any polygon, the area can be found using the vertices' coordinates with formulas related to determinants, like the one used here.