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The problem provided shows a geometric figure involving two triangles with parallel lines \( PQ \parallel TU \parallel SR \). We are given that \( SR = 54 \), \( TU = 10 \), and point \( T \) is the midpoint of \( SQ \), meaning that \( ST = TQ \). The goal is to determine the value of \( PQ \).

### Step-by-Step Solution:

We can use the concept of **similar triangles** because the lines \( PQ \), \( TU \), and \( SR \) are parallel, which means that triangle \( PQR \) is similar to triangle \( TUR \). The sides of these triangles are proportional.

From the figure, we have:

- \( PQ \parallel TU \parallel SR \) (given),
- \( ST = TQ \) (point \( T \) is the midpoint of \( SQ \)).

Since the triangles are similar, the ratio of corresponding sides is the same.

\[
\frac{PQ}{TU} = \frac{TU}{SR}
\]

Substitute the given values:

\[
\frac{PQ}{10} = \frac{10}{54}
\]

Now, solve for \( PQ \):

\[
PQ = \frac{10 \times 10}{54} = \frac{100}{54} \approx 1.85 \times 10 \approx 18.52
\]

This approximation is lower than expected for the options, so I need to recheck for alternative configurations or data interpretation.

Let me run this through again to cross-check

PQ, TU, and SR are parallel. If SR = 54, TU = 10, and T is the midpoint of SQ, find PQ. The given options are: (A) 34, (B) 32, (C) 28, (D) 26, (E) 24.

In this geometry problem, learn how to find the value of PQ given that PQ, TU, and SR are parallel, with SR = 54 and TU = 10. The solution involves the concept of similar triangles and proportionality.

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