https://file.aiquickdraw.com/mathbot/file/1726279381932r1ag6or4.jpg

The problem in the image asks to find the coordinates of point \( K \) such that \( JK \) is \(\frac{1}{9}\) of the total distance of \( JL \).

### Step-by-step solution:

1. **Coordinates of Points \( J \) and \( L \)**:
   - \( J = (-22, -32) \)
   - \( L = (5, 4) \)

2. **Find the vector \( \overrightarrow{JL} \)**:
   \[
   \overrightarrow{JL} = L - J = (5 - (-22), 4 - (-32)) = (5 + 22, 4 + 32) = (27, 36)
   \]

3. **To find point \( K \)** such that \( JK = \frac{1}{9} \times JL \), we need to move from point \( J \) towards point \( L \) by \( \frac{1}{9} \) of the vector \( \overrightarrow{JL} \).

   - Calculate \( \frac{1}{9} \times \overrightarrow{JL} \):
     \[
     \frac{1}{9} \times (27, 36) = \left( \frac{27}{9}, \frac{36}{9} \right) = (3, 4)
     \]

4. **Find the coordinates of \( K \)** by adding this vector to the coordinates of \( J \):
   \[
   K = J + \left( \frac{1}{9} \times \overrightarrow{JL} \right) = (-22, -32) + (3, 4) = (-22 + 3, -32 + 4) = (-19, -28)
   \]

Thus, the coordinates of point \( K \) are \( \boxed{(-19, -28)} \).

Would you like more details or have any questions?

Here are 5 related questions:
1. How is the vector between two points calculated?
2. What is the general formula for dividing a line segment in a given ratio?
3. How can you generalize this process to three-dimensional coordinates?
4. What if \( JK \) was \( \frac{1}{2} \) of \( JL \)? How would the solution change?
5. How is the distance between two points in the coordinate plane computed?

**Tip**: When working with line segments and ratios, always consider breaking down the steps into vectors and applying the ratio to get precise locations on the segment.

Find the coordinates of K so that JK is 1/9 of JL. Coordinates of J are (-22, -32) and coordinates of L are (5, 4).

Learn how to calculate the coordinates of point K such that JK is 1/9 of the total length of line segment JL. This problem involves using vector arithmetic and line segment division in coordinate geometry. Suitable for students in Grades 9-10.

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