Find the area of the triangle with the given vertices: A (12,4), B (1,1), C (0,8).

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.

Calculate the area of a triangle using the vertices A (12,4), B (1,1), and C (0,8) in a step-by-step solution. This problem applies coordinate geometry concepts and the triangle area formula, suitable for students in Grades 9-12.

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