The image shows a transformation of a triangle $\triangle PQR$ to $\triangle P'Q'R'$. To find the rule of translation, we need to determine how each point of the triangle is shifted from its original position to its new position.

Step-by-Step Analysis:

1. Identify the Coordinates:

   • Original Triangle:

     • $P$: (2, 5)
     • $Q$: (4, 7)
     • $R$: (0, 3)

   • Transformed Triangle:

     • $P'$: (2, -3)
     • $Q'$: (4, -1)
     • $R'$: (0, -5)

2. Determine the Translation:

   • Let's calculate the change in $x$- and $y$-coordinates from each point of $\triangle PQR$ to $\triangle P'Q'R'$.

   • $P$ to $P'$:

     • $x$-change: $2 - 2 = 0$
     • $y$-change: $-3 - 5 = -8$

   • $Q$ to $Q'$:

     • $x$-change: $4 - 4 = 0$
     • $y$-change: $-1 - 7 = -8$

   • $R$ to $R'$:

     • $x$-change: $0 - 0 = 0$
     • $y$-change: $-5 - 3 = -8$

3. Translation Rule: Each point is shifted 8 units down. Thus, the translation rule can be described as: $(x, y) \rightarrow (x, y - 8)$

Would you like to go through any more details or have questions about this translation?

Related Questions:

1. How would the translation change if the transformation moved the triangle 5 units to the right instead?
2. What would be the coordinates of $\triangle P'Q'R'$ if the translation were $(x, y) \rightarrow (x + 3, y - 8)$?
3. How does reflecting the triangle over the x-axis affect the coordinates of $\triangle PQR$?
4. What is the formula for a general translation in the coordinate plane?
5. Can you find the midpoint between $P$ and $P'$ in this translation?

Tip:

When working with translations, always subtract the original coordinates from the new ones to find the exact change in the $x$- and $y$-directions.