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.