Find the area of AABC, whose vertices are A(8, -4), B(3, 6) and C(-2, 4).

Answer

To find the area of triangle ABC with vertices A(8, -4), B(3, 6), and C(-2, 4), we can use the formula for the area of a triangle given its vertices in coordinate geometry.

### Step-by-Step Solution:

1. **Identify the coordinates of the vertices:**
- A = (8, -4) → (x1, y1)
- B = (3, 6) → (x2, y2)
- C = (-2, 4) → (x3, y3)

2. **Use the formula for the area of a triangle:**
The formula for the area \( A \) of a triangle with vertices at \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is given by:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

3. **Substitute the coordinates into the formula:**
- Here, \( x_1 = 8, y_1 = -4 \)
- \( x_2 = 3, y_2 = 6 \)
- \( x_3 = -2, y_3 = 4 \)

Plugging in these values:
\[ A = \frac{1}{2} \left| 8(6 - 4) + 3(4 - (-4)) + (-2)(-4 - 6) \right| \]

4. **Calculate the individual terms:**
- First term: \( 8(6 - 4) = 8 \times 2 = 16 \)
- Second term: \( 3(4 - (-4)) = 3 \times (4 + 4) = 3 \times 8 = 24 \)
- Third term: \( -2(-4 - 6) = -2 \times (-10) = 20 \)

5. **Combine the terms:**
\[ A = \frac{1}{2} \left| 16 + 24 + 20 \right| = \frac{1}{2} \left| 60 \right| = \frac{60}{2} = 30 \]

6. **Final result:**
The area of triangle ABC is \( 30 \) square units.