The area of the triangle formed by joining the points `(-5, 3)(6, -2) and (-3, 4)` is :

Options

A. 15
B. 30
C. 10.5
D. None of these

Answer

To find the area of the triangle formed by the points \((-5, 3)\), \((6, -2)\), and \((-3, 4)\), 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 1: Identify the coordinates
Let:
- \(A(x_1, y_1) = (-5, 3)\)
- \(B(x_2, y_2) = (6, -2)\)
- \(C(x_3, y_3) = (-3, 4)\)

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

\[ \text{Area} = \frac{1}{2} \left| (-5)((-2) - 4) + 6(4 - 3) + (-3)(3 - (-2)) \right| \]

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

1. Calculate \(y_2 - y_3 = -2 - 4 = -6\)
2. Calculate \(y_3 - y_1 = 4 - 3 = 1\)
3. Calculate \(y_1 - y_2 = 3 - (-2) = 5\)

Now substituting these values back into the expression:

\[ \text{Area} = \frac{1}{2} \left| (-5)(-6) + 6(1) + (-3)(5) \right| \]

Calculating each term:

- \((-5)(-6) = 30\)
- \(6(1) = 6\)
- \((-3)(5) = -15\)

Now substituting these values:

\[ \text{Area} = \frac{1}{2} \left| 30 + 6 - 15 \right| \]

### Step 4: Final calculation
Combine the terms inside the absolute value:

\[ 30 + 6 - 15 = 21 \]

So we have:

\[ \text{Area} = \frac{1}{2} \left| 21 \right| = \frac{21}{2} \]

### Final Answer
The area of the triangle is:

\[ \frac{21}{2} \text{ square units} \quad \text{or} \quad 10.5 \text{ square units} \]