https://r2.erweima.ai/mathbot/image/1717976266583nn3uk69j.png

To solve this problem, we need to determine the translation vector represented by the arrow and apply this translation to the vertices \( A \), \( B \), and \( C \) of triangle \( ABC \) to find the new vertices \( A' \), \( B' \), and \( C' \).

Let's break it down step by step:

1. **Determine the Translation Vector**:
   - Identify the start and end points of the arrow to find the translation vector.
   - From the image, the arrow starts at \((3, 6)\) and ends at \((5, 4)\).
   - The translation vector \( \vec{v} \) can be found by subtracting the coordinates of the start point from the end point: 
     \[
     \vec{v} = (5 - 3, 4 - 6) = (2, -2)
     \]

2. **Apply the Translation Vector**:
   - Apply the vector \( \vec{v} = (2, -2) \) to each vertex \( A \), \( B \), and \( C \).

3. **Calculate the New Coordinates**:
   - For vertex \( A \) at \((1, 5)\):
     \[
     A' = (1 + 2, 5 - 2) = (3, 3)
     \]
   - For vertex \( B \) at \((1, 3)\):
     \[
     B' = (1 + 2, 3 - 2) = (3, 1)
     \]
   - For vertex \( C \) at \((4, 3)\):
     \[
     C' = (4 + 2, 3 - 2) = (6, 1)
     \]

4. **Plot the New Points**:
   - \( A' \) is at \((3, 3)\)
   - \( B' \) is at \((3, 1)\)
   - \( C' \) is at \((6, 1)\)

This confirms the new triangle \( A'B'C' \) is correctly plotted in the image you provided.

Would you like further details on this solution or have any additional questions? Here are some questions you might consider asking:

1. How do I find the coordinates of a point after a translation?
2. What are the properties of translations in geometry?
3. How can I verify the correctness of a geometric transformation?
4. What other types of transformations can be applied to geometric shapes?
5. How do different transformations affect the coordinates of a shape?

**Tip:** Always double-check the coordinates of points after applying transformations to ensure accuracy.

Learn how to solve a translation problem in geometry step-by-step. This problem involves translating the vertices of a triangle using a given vector. Detailed explanation with examples provided. Suitable for high school students.

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