If the vertices of a triangle have the co-ordinates (0,0) , (3,2) and (0,5) then its area is

Options

A. `7 . 5`
B. ` 2 . 5`
C. 5
D. None of these

Answer

To find the area of the triangle with vertices at the coordinates (0,0), (3,2), and (0,5), we can use the formula for the area of a triangle given by 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-by-step Solution:

1. **Identify the coordinates of the vertices**:
- Let \( A(0, 0) \) be \( (x_1, y_1) \)
- Let \( B(3, 2) \) be \( (x_2, y_2) \)
- Let \( C(0, 5) \) be \( (x_3, y_3) \)

So, we have:
- \( x_1 = 0, y_1 = 0 \)
- \( x_2 = 3, y_2 = 2 \)
- \( x_3 = 0, y_3 = 5 \)

2. **Substitute the coordinates into the area formula**:
\[ \text{Area} = \frac{1}{2} \left| 0(2 - 5) + 3(5 - 0) + 0(0 - 2) \right| \]

3. **Calculate each term inside the absolute value**:
- The first term: \( 0(2 - 5) = 0 \)
- The second term: \( 3(5 - 0) = 15 \)
- The third term: \( 0(0 - 2) = 0 \)

Therefore, we have:
\[ \text{Area} = \frac{1}{2} \left| 0 + 15 + 0 \right| = \frac{1}{2} \left| 15 \right| \]

4. **Calculate the area**:
\[ \text{Area} = \frac{1}{2} \times 15 = 7.5 \]

Thus, the area of the triangle is **7.5 square units**.