The perimeter of a triangle with vertices (0,4),(0,0) and (3,0) is

Options

A. 5
B. 12
C. 11
D. `7+sqrt(5)`

Explanation

To find the perimeter of the triangle with vertices \( A(0, 4) \), \( O(0, 0) \), and \( B(3, 0) \), we will calculate the lengths of the sides \( AO \), \( OB \), and \( AB \) using the distance formula, and then sum these lengths to find the perimeter.

### Step-by-Step Solution:

1. **Identify the vertices of the triangle:**
- \( A(0, 4) \)
- \( O(0, 0) \)
- \( B(3, 0) \)

2. **Calculate the length of side \( AO \):**
- Using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
- For points \( A(0, 4) \) and \( O(0, 0) \):
\[ AO = \sqrt{(0 - 0)^2 + (0 - 4)^2} = \sqrt{0 + 16} = \sqrt{16} = 4 \]

3. **Calculate the length of side \( OB \):**
- For points \( O(0, 0) \) and \( B(3, 0) \):
\[ OB = \sqrt{(3 - 0)^2 + (0 - 0)^2} = \sqrt{9 + 0} = \sqrt{9} = 3 \]

4. **Calculate the length of side \( AB \):**
- For points \( A(0, 4) \) and \( B(3, 0) \):
\[ AB = \sqrt{(3 - 0)^2 + (0 - 4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

5. **Calculate the perimeter of the triangle:**
- The perimeter \( P \) is the sum of the lengths of the sides:
\[ P = AO + OB + AB = 4 + 3 + 5 = 12 \]

### Final Answer:
The perimeter of the triangle is \( 12 \) units.

---