what is the perimeter of the triangle with vertices
`A(-4, 2), B(0, -1)` and `C(3, 3)` ?

Options

A. `7+3sqrt(2)`
B. `10+5sqrt(2)`
C. `11+6sqrt(2)`
D. `5+sqrt(2)`

Explanation

To find the perimeter of the triangle with vertices A(-4, 2), B(0, -1), and C(3, 3), we will calculate the lengths of the sides AB, BC, and AC using the distance formula, and then sum these lengths to get the perimeter.

### Step 1: Calculate the length of side AB
The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
For points A(-4, 2) and B(0, -1):
- \(x_1 = -4\), \(y_1 = 2\)
- \(x_2 = 0\), \(y_2 = -1\)

Substituting these values into the distance formula:
\[ AB = \sqrt{(0 - (-4))^2 + (-1 - 2)^2} = \sqrt{(0 + 4)^2 + (-3)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 2: Calculate the length of side BC
For points B(0, -1) and C(3, 3):
- \(x_1 = 0\), \(y_1 = -1\)
- \(x_2 = 3\), \(y_2 = 3\)

Substituting these values into the distance formula:
\[ BC = \sqrt{(3 - 0)^2 + (3 - (-1))^2} = \sqrt{3^2 + (3 + 1)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 3: Calculate the length of side AC
For points A(-4, 2) and C(3, 3):
- \(x_1 = -4\), \(y_1 = 2\)
- \(x_2 = 3\), \(y_2 = 3\)

Substituting these values into the distance formula:
\[ AC = \sqrt{(3 - (-4))^2 + (3 - 2)^2} = \sqrt{(3 + 4)^2 + (3 - 2)^2} = \sqrt{7^2 + 1^2} = \sqrt{49 + 1} = \sqrt{50} = 5\sqrt{2} \]

### Step 4: Calculate the perimeter of the triangle
The perimeter \(P\) of the triangle is the sum of the lengths of its sides:
\[ P = AB + BC + AC = 5 + 5 + 5\sqrt{2} = 10 + 5\sqrt{2} \]

### Final Answer
The perimeter of the triangle is \(10 + 5\sqrt{2}\).

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