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The problem describes three points \( A = (1, -5, k) \), \( B = (3, k, 1) \), and \( C = (k, -5, 2) \), which are the vertices of triangle \( ABC \). We are asked to determine the value of \( k \) so that the triangle is right-angled at point \( A \).

### Steps to solve:
For the triangle to be right-angled at \( A \), the vectors \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \) must be orthogonal. This means the dot product of these two vectors must be zero.

1. **Find vectors**:
   \[
   \overrightarrow{AB} = B - A = (3 - 1, k - (-5), 1 - k) = (2, k + 5, 1 - k)
   \]
   \[
   \overrightarrow{AC} = C - A = (k - 1, -5 - (-5), 2 - k) = (k - 1, 0, 2 - k)
   \]

2. **Dot product condition**:
   For orthogonal vectors, the dot product \( \overrightarrow{AB} \cdot \overrightarrow{AC} = 0 \). Thus, we calculate:
   \[
   \overrightarrow{AB} \cdot \overrightarrow{AC} = (2)(k - 1) + (k + 5)(0) + (1 - k)(2 - k)
   \]
   Simplify the expression:
   \[
   2(k - 1) + (1 - k)(2 - k) = 0
   \]
   Expand and solve for \( k \):
   \[
   2k - 2 + (1 - k)(2 - k) = 0
   \]
   \[
   2k - 2 + (2 - k - 2k + k^2) = 0
   \]
   \[
   2k - 2 + 2 - k - 2k + k^2 = 0
   \]
   \[
   k^2 - k = 0
   \]
   Factor the quadratic equation:
   \[
   k(k - 1) = 0
   \]
   Therefore, \( k = 0 \) or \( k = 1 \).

### Conclusion:
The value of \( k \) can be either 0 or 1 for the triangle to be right-angled at point \( A \).

Do you want any further details or clarifications?

### Related Questions:
1. How do you compute the length of a vector in 3D space?
2. What is the geometric interpretation of the dot product in vector analysis?
3. How do you solve quadratic equations by factoring?
4. What other conditions, besides orthogonality, are necessary for a triangle to be right-angled?
5. Can a triangle have multiple right angles at different vertices?

### Tip:
When working with vectors, always check if they are perpendicular by verifying that their dot product is zero.

Sean los puntos A = (1, -5, k), B = (3, k, 1) y C = (k, -5, 2) tres vértices de un triángulo ABC. Determine el valor de k para que el triángulo sea recto en A.

Solve for the value of k such that the triangle ABC, with vertices A(1, -5, k), B(3, k, 1), and C(k, -5, 2), is right-angled at point A. This involves using the dot product of vectors and solving a quadratic equation. Suitable for students in grades 10-12.

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