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The problem asks to calculate the perimeter of a triangle with vertices at points \( A(0,4) \), \( B(4,2) \), and \( C(3,1) \), and determine if it is a right triangle. Let's work through this step by step:

### Step 1: Calculate the lengths of the sides
The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

- **Length of side \( AB \):**
  \[
  AB = \sqrt{(4 - 0)^2 + (2 - 4)^2} = \sqrt{4^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}
  \]

- **Length of side \( BC \):**
  \[
  BC = \sqrt{(3 - 4)^2 + (1 - 2)^2} = \sqrt{(-1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}
  \]

- **Length of side \( AC \):**
  \[
  AC = \sqrt{(3 - 0)^2 + (1 - 4)^2} = \sqrt{3^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
  \]

### Step 2: Calculate the perimeter
The perimeter \( P \) is the sum of all sides:

\[
P = AB + BC + AC = 2\sqrt{5} + \sqrt{2} + 3\sqrt{2}
\]

Simplifying \( BC + AC \):

\[
\sqrt{2} + 3\sqrt{2} = 4\sqrt{2}
\]

Thus, the perimeter is:

\[
P = 2\sqrt{5} + 4\sqrt{2}
\]

### Step 3: Check if it is a right triangle
A triangle is a right triangle if the Pythagorean theorem holds, meaning the square of the longest side should be equal to the sum of the squares of the other two sides.

- \( AB^2 = (2\sqrt{5})^2 = 20 \)
- \( BC^2 = (\sqrt{2})^2 = 2 \)
- \( AC^2 = (3\sqrt{2})^2 = 18 \)

Check if \( AB^2 + BC^2 = AC^2 \):

\[
AB^2 + BC^2 = 20 + 2 = 22
\]

Since \( 22 \neq 18 \), the triangle is **not a right triangle**.

### Summary:
- The perimeter of the triangle is \( 2\sqrt{5} + 4\sqrt{2} \).
- The triangle is not a right triangle.

Do you have any questions or want more details? Here are some related questions:

1. How do you simplify square roots in perimeter calculations?
2. How can you identify the longest side in a triangle to apply the Pythagorean theorem?
3. What are some alternative methods for verifying if a triangle is right-angled?
4. How does the distance formula relate to finding the perimeter of geometric shapes?
5. Can you explain how triangles are classified based on angles?

**Tip:** When calculating the perimeter of geometric shapes, always ensure the units of distance are consistent across all sides!

Calcola il perimetro del triangolo che ha i vertici nei punti di coordinate A(0; 4), B(4; 2) e C(3; 1). Si tratta di un triangolo rettangolo? Motiva la risposta.

Calculate the perimeter of the triangle with vertices A(0; 4), B(4; 2), and C(3; 1), and determine if it is a right triangle. Learn how to apply the distance formula and verify using the Pythagorean Theorem.

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