The diagram shows two triangles, $\triangle QRS$ and $\triangle PRT$, where $\overline{QS} \parallel \overline{PT}$, and the lengths of segments $\overline{QR} = 3$, $\overline{PQ} = 6$, and the perimeter of $\triangle QRS = 11$.

To find the perimeter of $\triangle PRT$, we will use the fact that $\overline{QS} \parallel \overline{PT}$, meaning the triangles $\triangle QRS$ and $\triangle PRT$ are similar by AA similarity (since they share angle $R$ and have parallel sides).

Step 1: Ratio of Similarity

Since $\overline{QS} \parallel \overline{PT}$, the ratio of the sides in the two triangles is proportional. The ratio of corresponding sides is: $\text{Ratio} = \frac{PQ}{QR} = \frac{6}{3} = 2.$ Thus, all sides of $\triangle PRT$ are double the corresponding sides of $\triangle QRS$.

Step 2: Perimeter of $\triangle PRT$

Since the perimeter of $\triangle QRS$ is given as 11 units, and the triangles are similar with a ratio of 2, the perimeter of $\triangle PRT$ is: $\text{Perimeter of } \triangle PRT = 2 \times \text{Perimeter of } \triangle QRS = 2 \times 11 = 22 \, \text{units}.$

Final Answer:

The perimeter of $\triangle PRT$ is $\boxed{22}$ units.

Do you want more details or have any questions? Here are 5 questions to expand on this:

1. How do we confirm that the triangles are similar?
2. Why is the ratio of the perimeters the same as the ratio of the sides in similar triangles?
3. What happens if the given perimeter of $\triangle QRS$ was different?
4. Can we calculate the exact lengths of $\overline{RS}$ and $\overline{RT}$?
5. How would the problem change if $\overline{QS}$ and $\overline{PT}$ were not parallel?

Tip: In similar triangles, the ratio of corresponding sides is constant, which also applies to their perimeters and areas.