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### Question 6:
- **Translation Vector from \( J \) to \( J' \)**:
  - The coordinates of \( J \) are \( (6, 4) \), and the coordinates of \( J' \) are \( (-6, -4) \).
  - The translation vector \( \vec{v} \) is given by:
    \[
    \vec{v} = (x_2 - x_1, y_2 - y_1) = (-6 - 6, -4 - 4) = (-12, -8)
    \]
  - Therefore, the translation vector is \( \vec{v} = (-12, -8) \).

### Question 7:
- **Slope of the vector \( \vec{v} \)**:
  - The slope of a vector \( \vec{v} = (x, y) \) is given by:
    \[
    \text{slope} = \frac{\Delta y}{\Delta x} = \frac{-8}{-12} = \frac{2}{3}
    \]
  - So, the slope of the vector is \( \frac{2}{3} \).

### Question 8:
- **Coordinates of \( D, E, F \) after translation**:
  - Given that triangle \( J'K'L' \) is also the image of \( \triangle DEF \) under a translation along the vector \( (-6, 4) \):
    - To find the coordinates of \( D, E, F \), subtract the translation vector from each point of \( J'K'L' \).

  - Coordinates of \( J'(-6, -4) \):
    - \( D = (-6 - (-6), -4 - 4) = (0, -8) \)

  - Coordinates of \( K'(-2, 4) \):
    - \( E = (-2 - (-6), 4 - 4) = (4, 0) \)

  - Coordinates of \( L'(-8, -6) \):
    - \( F = (-8 - (-6), -6 - 4) = (-2, -10) \)

Would you like to see any further explanations or details on these problems?

### Here are five related questions:
1. How do we calculate the translation vector if both triangles were rotated instead of translated?
2. What would be the new coordinates if the triangle \( JKL \) was translated by a different vector, say \( (5, -3) \)?
3. How can you calculate the distance between the original and translated points geometrically?
4. How do transformations like reflection or rotation differ from translations in terms of vector usage?
5. How can the slope of the vector \( \vec{v} \) be used to describe the direction of movement?

### Tip:
Always ensure that the direction of a translation vector is consistent by subtracting the correct initial and final coordinates in order.

6. Triangle J'K'L' is the image of ΔJKL under a translation. Draw the translation vector v from J to its image in ΔJ'K'L'. Write the vector in component form. 7. What is the slope of v? 8. Triangle J'K'L' is also the image of ΔDEF under a translation along a vector (-6, 4). Find the coordinates of points D, E, and F, and draw ΔDEF.

Learn how to find the translation vector and slope for triangle JKL to J'K'L' and calculate the coordinates of triangle DEF under a translation along a vector. Includes step-by-step solutions and concepts in coordinate geometry and vector operations. Suitable for students in Grades 8-10.

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