In the image, we see two triangles: the original triangle $TUV$ in black, and its image $T'U'V'$ after a translation in blue. The goal is to determine the translation rule that maps triangle $TUV$ to triangle $T'U'V'$.

Step 1: Choose a Point

To identify the translation, we can select one point from each triangle and observe how the point moved. Let's use point $T$ (in black) and its translated point $T'$ (in blue).

Step 2: Measure the Translation

• The coordinates of $T$ are approximately $(1, 4)$.
• The coordinates of $T'$ are approximately $(-4, 2)$.

Now, calculate the movement:

• The x-coordinate moved from 1 to -4, which is a shift of 5 units left (since $1 - (-4) = 5$).
• The y-coordinate moved from 4 to 2, which is a shift of 2 units down (since $4 - 2 = 2$).

Step 3: Write the Rule

The translation can be described as 5 units left and 2 units down.

Looking at the answer choices:

• Option c: 5 units left and 2 units up is closest, but it says "up" instead of "down."
• None of the provided options exactly match the translation rule based on the observed shifts.

Would you like more details or further clarification on the problem? Here are five related questions to expand your understanding:

1. How can you verify the translation by checking other points like $U$ and $V$?
2. What is the general form of a translation rule in coordinate geometry?
3. How does translation differ from other transformations like rotation or reflection?
4. How can you confirm a translation by comparing the lengths of sides and angles of the original and translated figures?
5. Why do translations not change the orientation of a figure?

Tip: When analyzing translations, always check both the x- and y-coordinates for shifts to avoid confusion with other transformations like reflections.