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We are given the coordinates of the vertices of triangle \( ABC \) in 3D space:

- \( A = (1, 0, 0) \)
- \( B = (0, 1, 0) \)
- \( C = (0, 0, 1) \)

### 1. **Area of the triangle:**

To find the area of a triangle in 3D, we can use the formula:

\[
\text{Area} = \frac{1}{2} \times \|\mathbf{AB} \times \mathbf{AC}\|
\]

Where:
- \( \mathbf{AB} = B - A = (0 - 1, 1 - 0, 0 - 0) = (-1, 1, 0) \)
- \( \mathbf{AC} = C - A = (0 - 1, 0 - 0, 1 - 0) = (-1, 0, 1) \)

Now, let's calculate the cross product \( \mathbf{AB} \times \mathbf{AC} \).

### 2. **Perimeter of the triangle:**

The perimeter is the sum of the lengths of the three sides of the triangle. The lengths of the sides can be found using the distance formula:

\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]

We need to find the lengths of:
- \( AB \)
- \( BC \)
- \( AC \)

Let me calculate these values.The area of the triangle is \( \frac{\sqrt{3}}{2} \) (approximately 0.866), and the perimeter is \( 2\sqrt{2} + \sqrt{2} \) (approximately 4.242).

Based on these values, the correct answer is:

- **Area:** \( \frac{\sqrt{3}}{2} \)
- **Perimeter:** \( 2\sqrt{2} \)

This matches option **C**.

Would you like more details on any part of the solution, or to discuss related topics? Here are some related questions you may find helpful:

1. How do we derive the formula for the area of a triangle in 3D space?
2. Can we calculate the area of a triangle differently, using Heron's formula in 3D?
3. How does the cross product work in vector mathematics?
4. What are other ways to compute perimeter in 3D geometry?
5. How is the concept of perimeter extended to polygons in higher dimensions?

**Tip:** When working with vectors, cross products give perpendicular vectors, and their magnitudes can help find areas of geometric shapes.

El triángulo ABC de la figura 11 tiene sus vértices ubicados en las coordenadas A = (1, 0, 0), B = (0, 1, 0) y C = (0, 0, 1). Su área y su perímetro miden, respectivamente, ¿cuál es la opción correcta?

Resuelve el área y el perímetro de un triángulo en 3D cuyas coordenadas son A(1, 0, 0), B(0, 1, 0) y C(0, 0, 1) usando productos cruzados y fórmulas de distancia. Adecuado para estudiantes de secundaria avanzada (Grados 10-12).

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