Find the area of that triangle whose vertices are `(-4,3),(-2,1)and(5,2).`

Answer

To find the area of the triangle with vertices at \((-4, 3)\), \((-2, 1)\), and \((5, 2)\), we can use the formula for the area of a triangle given its vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):

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

### Step-by-Step Solution:

1. **Identify the coordinates**:
- Let \((x_1, y_1) = (-4, 3)\)
- Let \((x_2, y_2) = (-2, 1)\)
- Let \((x_3, y_3) = (5, 2)\)

2. **Substitute the values into the area formula**:
\[ \text{Area} = \frac{1}{2} \left| -4(1 - 2) + (-2)(2 - 3) + 5(3 - 1) \right| \]

3. **Calculate each term**:
- For the first term:
\[ -4(1 - 2) = -4(-1) = 4 \]
- For the second term:
\[ -2(2 - 3) = -2(-1) = 2 \]
- For the third term:
\[ 5(3 - 1) = 5(2) = 10 \]

4. **Combine the results**:
\[ \text{Area} = \frac{1}{2} \left| 4 + 2 + 10 \right| = \frac{1}{2} \left| 16 \right| = \frac{16}{2} = 8 \]

5. **Final result**:
The area of the triangle is \(8\) square units.