The perimeter of a triangle with vertices A(0, 4), B(0, 0) and C(3, 0):

Options

A. 3
B. 5
C. 10
D. 12

Answer

To find the perimeter of the triangle with vertices A(0, 4), B(0, 0), and C(3, 0), we will calculate the lengths of the sides AB, BC, and CA using the distance formula, and then sum these lengths.

### Step 1: Calculate the length of side AB
The coordinates of points A and B are:
- A(0, 4)
- B(0, 0)

Using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of A and B:
\[ AB = \sqrt{(0 - 0)^2 + (0 - 4)^2} = \sqrt{0 + 16} = \sqrt{16} = 4 \]

### Step 2: Calculate the length of side BC
The coordinates of points B and C are:
- B(0, 0)
- C(3, 0)

Using the distance formula:
\[ BC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of B and C:
\[ BC = \sqrt{(3 - 0)^2 + (0 - 0)^2} = \sqrt{9 + 0} = \sqrt{9} = 3 \]

### Step 3: Calculate the length of side CA
The coordinates of points C and A are:
- C(3, 0)
- A(0, 4)

Using the distance formula:
\[ CA = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of C and A:
\[ CA = \sqrt{(0 - 3)^2 + (4 - 0)^2} = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 4: Calculate the perimeter of the triangle
Now, we can find the perimeter (P) by summing the lengths of all sides:
\[ P = AB + BC + CA = 4 + 3 + 5 = 12 \]

### Final Answer
The perimeter of the triangle is **12**.

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