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The problem involves finding the coordinates of a point \( K \) that lies on the line segment \( JL \) such that the ratio \( \frac{JK}{KL} = \frac{1}{3} \).

### Step 1: Identify the coordinates of points \( J \) and \( L \).

- Coordinates of \( J \) are \( (-18, 17) \).
- Coordinates of \( L \) are \( (6, -13) \).

### Step 2: Use the section formula.
The section formula is used to find a point \( K(x, y) \) that divides the line segment joining two points \( J(x_1, y_1) \) and \( L(x_2, y_2) \) in a given ratio \( m:n \). The coordinates of \( K \) are given by:

\[
K(x, y) = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)
\]

Here, \( m = 1 \) and \( n = 3 \).

### Step 3: Apply the section formula.

- For the \( x \)-coordinate:
\[
x_K = \frac{1(6) + 3(-18)}{1 + 3} = \frac{6 - 54}{4} = \frac{-48}{4} = -12
\]

- For the \( y \)-coordinate:
\[
y_K = \frac{1(-13) + 3(17)}{1 + 3} = \frac{-13 + 51}{4} = \frac{38}{4} = 9.5
\]

### Final Answer:
The coordinates of \( K \) are \( (-12, 9.5) \).

Do you want more details or have any questions?

Here are some related questions:
1. How does the section formula change if \( \frac{JK}{KL} \) was instead \( \frac{2}{3} \)?
2. What if the problem asked for the midpoint instead of dividing the segment in a specific ratio?
3. Can you generalize the section formula for any ratio \( m:n \)?
4. How does the distance formula verify the correctness of this solution?
5. What happens if the coordinates of \( J \) or \( L \) were negative or non-integer?

**Tip:** Always double-check the ratio given in the problem and apply it carefully to the section formula to avoid mistakes in calculations.

Learn how to find the coordinates of point K on the line segment JL, dividing it in the ratio 1:3. This problem involves using the section formula in coordinate geometry with detailed step-by-step calculations.

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