Find the area of triangle whose vertices are `(-15,3),(6,-2) and (-3,4)`.

Answer

To find the area of the triangle with vertices at the points \((-15, 3)\), \((6, -2)\), and \((-3, 4)\), we can use the formula for the area of a triangle given its vertices:

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

### Step 1: Assign the coordinates
Let:
- \( (x_1, y_1) = (-15, 3) \)
- \( (x_2, y_2) = (6, -2) \)
- \( (x_3, y_3) = (-3, 4) \)

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

\[ \text{Area} = \frac{1}{2} \left| -15((-2) - 4) + 6(4 - 3) + (-3)(3 - (-2)) \right| \]

### Step 3: Simplify the expression
Now, simplify each term inside the absolute value:

1. Calculate \(y_2 - y_3\):
\[ -2 - 4 = -6 \]
So, the first term becomes:
\[ -15 \times -6 = 90 \]

2. Calculate \(y_3 - y_1\):
\[ 4 - 3 = 1 \]
So, the second term becomes:
\[ 6 \times 1 = 6 \]

3. Calculate \(y_1 - y_2\):
\[ 3 - (-2) = 3 + 2 = 5 \]
So, the third term becomes:
\[ -3 \times 5 = -15 \]

### Step 4: Combine the terms
Now, combine the results:
\[ \text{Area} = \frac{1}{2} \left| 90 + 6 - 15 \right| \]
\[ = \frac{1}{2} \left| 81 \right| \]

### Step 5: Calculate the final area
Finally, calculate the area:
\[ \text{Area} = \frac{1}{2} \times 81 = 40.5 \]

Thus, the area of the triangle is \( 40.5 \) square units.

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