The area of the triangle with vertices at (-4,1),(1,2),(4,-3) is

Options

A. 14
B. 16
C. 15
D. None of these

Answer

To find the area of the triangle with vertices at points A(-4, 1), B(1, 2), and C(4, -3), we can use the formula for the area of a triangle given by the coordinates of its vertices:

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

Where \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) are the coordinates of the vertices.

### Step 1: Assign the coordinates
Let:
- \(A(-4, 1) \Rightarrow (x_1, y_1) = (-4, 1)\)
- \(B(1, 2) \Rightarrow (x_2, y_2) = (1, 2)\)
- \(C(4, -3) \Rightarrow (x_3, y_3) = (4, -3)\)

### Step 2: Substitute the coordinates into the area formula
Substituting the coordinates into the area formula:

\[ \text{Area} = \frac{1}{2} \left| -4(2 - (-3)) + 1((-3) - 1) + 4(1 - 2) \right| \]

### Step 3: Simplify the expression
Calculating each term step-by-step:

1. Calculate \(2 - (-3) = 2 + 3 = 5\)
2. Calculate \((-3) - 1 = -4\)
3. Calculate \(1 - 2 = -1\)

Now substituting these values back into the area formula:

\[ \text{Area} = \frac{1}{2} \left| -4(5) + 1(-4) + 4(-1) \right| \]

### Step 4: Calculate the products
Calculating the products:

1. \(-4 \times 5 = -20\)
2. \(1 \times -4 = -4\)
3. \(4 \times -1 = -4\)

Now substituting these results:

\[ \text{Area} = \frac{1}{2} \left| -20 - 4 - 4 \right| \]

### Step 5: Combine the terms
Combine the terms inside the absolute value:

\[ -20 - 4 - 4 = -28 \]

So,

\[ \text{Area} = \frac{1}{2} \left| -28 \right| = \frac{1}{2} \times 28 = 14 \]

### Conclusion
Thus, the area of the triangle is \(14\) square units.