To find $x$ from the given image, we need to use the fact that the two triangles $\triangle PQR$ and $\triangle SQT$ are similar. This is because they have corresponding equal angles ($\angle P = \angle S$, $\angle Q = \angle Q$, and $\angle R = \angle T$), which gives us a condition of similarity by AA (Angle-Angle).

Step-by-step approach:

1. Similarity of triangles: Since $\triangle PQR \sim \triangle SQT$, the corresponding sides of these triangles are proportional. This gives us the relation:

   $\frac{PQ}{QS} = \frac{QR}{QT} = \frac{PR}{ST}$

2. Given information: From the diagram:

   • $PQ = 9$
   • $QS = 3$
   • $QR = 13.5$
   • $QT = 3x$
   • $PR = 28^\circ$
   • $ST = 3x$

3. Using the proportion: Using the proportion of corresponding sides from the similar triangles:

   $\frac{PQ}{QS} = \frac{QR}{QT}$

   Substituting the given values:

   $\frac{9}{3} = \frac{13.5}{3x}$

   Simplifying:

   $3 = \frac{13.5}{3x}$

   To solve for $x$, multiply both sides by $3x$:

   $3(3x) = 13.5$

   Simplifying:

   $9x = 13.5$

   Solving for $x$:

   $x = \frac{13.5}{9} = 1.5$

Thus, the value of $x$ is $\boxed{1.5}$.

Would you like further details on this process?

Here are some related questions:

1. How do you confirm triangle similarity in geometry?
2. What are other properties of similar triangles?
3. How can proportionality help in solving real-world problems?
4. Can you solve for unknown angles using triangle similarity?
5. What are some other methods for finding missing sides in triangles?

Tip: Always ensure you check whether triangles are similar using the AA, SSS, or SAS criteria before applying proportions.