Find the value of K for which the points (8,1), (k , - 4), (2, -5) are collinear.

Answer

To find the value of \( k \) for which the points \( (8, 1) \), \( (k, -4) \), and \( (2, -5) \) are collinear, we can use the concept of slopes. The points are collinear if the slope between any two pairs of points is the same.

### Step-by-Step Solution:

1. **Identify the points**:
- Let \( A = (8, 1) \)
- Let \( B = (k, -4) \)
- Let \( C = (2, -5) \)

2. **Calculate the slope of line segment AB**:
The formula for the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
For points \( A \) and \( B \):
\[ \text{slope of } AB = \frac{-4 - 1}{k - 8} = \frac{-5}{k - 8} \]

3. **Calculate the slope of line segment BC**:
For points \( B \) and \( C \):
\[ \text{slope of } BC = \frac{-5 - (-4)}{2 - k} = \frac{-5 + 4}{2 - k} = \frac{-1}{2 - k} \]

4. **Set the slopes equal to each other**:
Since the points are collinear, we set the slopes equal:
\[ \frac{-5}{k - 8} = \frac{-1}{2 - k} \]

5. **Cross-multiply to solve for \( k \)**:
Cross-multiplying gives:
\[ -5(2 - k) = -1(k - 8) \]
Expanding both sides:
\[ -10 + 5k = -k + 8 \]

6. **Rearrange the equation**:
Bring all terms involving \( k \) to one side and constant terms to the other side:
\[ 5k + k = 8 + 10 \]
\[ 6k = 18 \]

7. **Solve for \( k \)**:
Divide both sides by 6:
\[ k = \frac{18}{6} = 3 \]

### Final Answer:
The value of \( k \) for which the points are collinear is \( k = 3 \).