The area of the triangle whose vertices are given by `(1,2), (-4,-3)` and `(4,1)` is :

Options

A. `7` sq. units
B. `20` sq. units
C. `10` sq. units
D. `14` sq. units

Answer

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

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

3. **Calculate each term inside the absolute value:**
- First term: \( 1((-3) - 1) = 1 \times (-4) = -4 \)
- Second term: \( -4(1 - 2) = -4 \times (-1) = 4 \)
- Third term: \( 4(2 - (-3)) = 4 \times (2 + 3) = 4 \times 5 = 20 \)

4. **Combine the terms:**
\[ \text{Area} = \frac{1}{2} \left| -4 + 4 + 20 \right| = \frac{1}{2} \left| 20 \right| = \frac{20}{2} = 10 \]

5. **Final result:**
The area of the triangle is \( 10 \) square units.