To solve this problem, let's break down each component step-by-step.

Problem Analysis

We are given:

1. Triangle PQR with vertices $P(5,1)$, $Q(2,2)$, and $R(3,4)$.
2. Two transformations:
   • $r_{b}$: Reflection over the line $y = 8$.
   • $r_{x\text{-axis}}$: Reflection over the x-axis.

We need to find a translation that is equivalent to the composition of these transformations, $r_{b} \circ r_{x\text{-axis}}$, and express it in terms of a translation $T$ that maps $\triangle PQR$ accordingly.

Step-by-Step Solution

1. Reflect over the x-axis:

   • Reflecting over the x-axis changes the $y$-coordinates of the points to their negatives.
   • Thus:
     • $P(5,1) \rightarrow P'(5,-1)$
     • $Q(2,2) \rightarrow Q'(2,-2)$
     • $R(3,4) \rightarrow R'(3,-4)$

2. Reflect over the line $y = 8$:

   • To reflect over the line $y = 8$, calculate the new positions of $P'$, $Q'$, and $R'$ by using the fact that the reflection of a point $(x, y)$ over the line $y = 8$ will be $(x, 16 - y)$.
   • Applying this:
     • $P'(5, -1) \rightarrow P''(5, 16 - (-1)) = P''(5, 17)$
     • $Q'(2, -2) \rightarrow Q''(2, 16 - (-2)) = Q''(2, 18)$
     • $R'(3, -4) \rightarrow R''(3, 16 - (-4)) = R''(3, 20)$

3. Determine the Translation:

   • Now we compare the original points $P(5,1)$, $Q(2,2)$, and $R(3,4)$ with the final points $P''(5,17)$, $Q''(2,18)$, and $R''(3,20)$.
   • Observing the change:
     • $P$ moved from $(5,1)$ to $(5,17)$, a vertical shift of $+16$.
     • $Q$ moved from $(2,2)$ to $(2,18)$, a vertical shift of $+16$.
     • $R$ moved from $(3,4)$ to $(3,20)$, a vertical shift of $+16$.

   Therefore, the translation that is equivalent to the composition of these reflections is a vertical translation of 16 units.

Final Answer

The equivalent translation $T$ is: $T(0, 16)$

Would you like further details or have any questions?

Here are some related questions that could extend your understanding:

1. How would the equivalent translation change if the line $b$ was at $y = 10$?
2. What would happen if the reflection sequence was reversed, i.e., $r_{x\text{-axis}} \circ r_{b}$?
3. How can you generalize the translation vector for any line $y = c$ instead of $y = 8$?
4. What are other types of transformations that are equivalent to compositions of reflections?
5. How can this method be applied in 3D geometry?

Tip: Reflections over lines other than the axes can often be converted into translations when combined with additional reflections.