Find the area of the triangle whose vertices are (- 4, 8), (6, - 6) and (-3, -2).

Answer

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

### Step-by-Step Solution:

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

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

3. **Substitute the coordinates into the formula**:
Plugging in the coordinates:
\[ A = \frac{1}{2} \left| -4(-6 - (-2)) + 6(-2 - 8) + (-3)(8 - (-6)) \right| \]
Simplifying each term:
- For the first term:
\[ -4(-6 + 2) = -4(-4) = 16 \]
- For the second term:
\[ 6(-2 - 8) = 6(-10) = -60 \]
- For the third term:
\[ -3(8 + 6) = -3(14) = -42 \]

4. **Combine the terms**:
Now, we can combine these results:
\[ A = \frac{1}{2} \left| 16 - 60 - 42 \right| \]
Simplifying inside the absolute value:
\[ 16 - 60 - 42 = 16 - 102 = -86 \]
Thus, we have:
\[ A = \frac{1}{2} \left| -86 \right| = \frac{1}{2} \times 86 = 43 \]

5. **Final result**:
Therefore, the area of the triangle is \( 43 \) square units.