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The problem asks to determine the value of \( k \) such that triangle \( ABC \) is a right triangle with a right angle at point \( A \), where the points are given as:

- \( A = (1, -5, k) \)
- \( B = (3, k, 1) \)
- \( C = (k, -5, 2) \)

Since the triangle is right-angled at point \( A \), the vectors \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \) must be perpendicular. Two vectors are perpendicular if their dot product equals zero.

### Step 1: Find vectors \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \)
- \( \overrightarrow{AB} = B - A = (3 - 1, k - (-5), 1 - k) = (2, k + 5, 1 - k) \)
- \( \overrightarrow{AC} = C - A = (k - 1, -5 - (-5), 2 - k) = (k - 1, 0, 2 - k) \)

### Step 2: Dot product of \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \)
The dot product of two vectors \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) is given by:
\[
\overrightarrow{AB} \cdot \overrightarrow{AC} = x_1 \cdot x_2 + y_1 \cdot y_2 + z_1 \cdot z_2
\]
For \( \overrightarrow{AB} = (2, k + 5, 1 - k) \) and \( \overrightarrow{AC} = (k - 1, 0, 2 - k) \), the dot product is:
\[
\overrightarrow{AB} \cdot \overrightarrow{AC} = 2 \cdot (k - 1) + (k + 5) \cdot 0 + (1 - k) \cdot (2 - k)
\]
Simplifying this expression:
\[
= 2(k - 1) + (1 - k)(2 - k)
= 2k - 2 + (1 - k)(2 - k)
\]
Expand \( (1 - k)(2 - k) \):
\[
(1 - k)(2 - k) = 2 - k - 2k + k^2 = k^2 - 3k + 2
\]
Now, the dot product becomes:
\[
2k - 2 + k^2 - 3k + 2 = k^2 - k
\]

### Step 3: Set the dot product equal to zero
Since the vectors are perpendicular:
\[
k^2 - k = 0
\]
Factor the equation:
\[
k(k - 1) = 0
\]

### Step 4: Solve for \( k \)
From \( k(k - 1) = 0 \), the solutions are:
\[
k = 0 \quad \text{or} \quad k = 1
\]

Thus, the values of \( k \) that make the triangle right-angled at \( A \) are \( k = 0 \) and \( k = 1 \).

Let me know if you need further clarifications or want more details! Here are some related questions for deeper understanding:

1. How can we check if the vectors are perpendicular by using the cross product?
2. How would the solution change if the right angle were at a different vertex?
3. What does it mean geometrically for two vectors to be perpendicular?
4. Can we apply this method to triangles in two dimensions?
5. How does the dot product help in identifying orthogonal vectors in general?

**Tip:** Always check the geometric properties (like perpendicularity or parallelism) using vector operations, as they provide reliable results when working in multi-dimensional spaces.

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.

This problem involves using vector algebra to find the value of k such that triangle ABC has a right angle at point A. Concepts include vector dot products, perpendicularity conditions, and solving a quadratic equation. Suitable for students in Grades 10-12.

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