https://file.aiquickdraw.com/mathbot/file/1730469462075y8we56ha.jpg

Let's go through each question step by step:

1. **Name the vector and write its component form:**
   - The vector in the image is **vector KB**.
   - To find the component form of a vector, use the coordinates of the starting and ending points. For vector \( \overrightarrow{KB} \):
     - \( K \) is located at \( (1, 4) \) and \( B \) at \( (5, 2) \).
     - Component form: \( \overrightarrow{KB} = (5 - 1, 2 - 4) = (4, -2) \).

2. **Translate \( \triangle LMN \) using the vector \( \langle -2, 6 \rangle \):**
   - Original vertices:
     - \( L(2, 2) \)
     - \( M(5, 3) \)
     - \( N(9, 1) \)
   - Translation rule: \( (x, y) \rightarrow (x - 2, y + 6) \)
     - New \( L' \): \( (2 - 2, 2 + 6) = (0, 8) \)
     - New \( M' \): \( (5 - 2, 3 + 6) = (3, 9) \)
     - New \( N' \): \( (9 - 2, 1 + 6) = (7, 7) \)

3. **Write a rule to translate \( \triangle A'B'C' \) back to \( \triangle ABC \):**
   - The translation rule will be the inverse of whatever translation was applied in Example 3. For instance, if Example 3 translated \( \triangle ABC \) by \( (x + a, y + b) \), then translating back would be \( (x - a, y - b) \).

4. **Graph \( \triangle RST \) and its image after translation:**
   - Original vertices:
     - \( R(2, 2) \)
     - \( S(5, 2) \)
     - \( T(3, 5) \)
   - Translation rule: \( (x, y) \rightarrow (x + 1, y + 2) \)
     - New \( R' \): \( (2 + 1, 2 + 2) = (3, 4) \)
     - New \( S' \): \( (5 + 1, 2 + 2) = (6, 4) \)
     - New \( T' \): \( (3 + 1, 5 + 2) = (4, 7) \)

Would you like more details or have any questions?

### Related Questions

1. What is the difference between translating a shape and reflecting it?
2. How can you determine the component form of a vector if only the vector's direction is known?
3. What is an example of a translation that results in a reflection along a line?
4. How would you rotate \( \triangle RST \) instead of translating it?
5. Can multiple vectors achieve the same translation? Explain with examples.

**Tip:** When translating a shape, make sure to apply the same transformation to each vertex.

Monitoring Progress Questions: Translate and identify vectors for given points and triangles.

Detailed solutions for translating triangles and finding vector components in coordinate geometry. Includes step-by-step explanations for vector KB, translating triangle LMN, reversing triangle A'B'C', and translating triangle RST.

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