We are given that $PQ \parallel RT$, and we need to find $RS$ using the Triangle Proportionality Theorem.

Step 1: Understanding the Theorem

The Triangle Proportionality Theorem states that if a line parallel to one side of a triangle intersects the other two sides, it divides those sides proportionally.

Thus, we set up the proportion: $\frac{RS}{ST} = \frac{RQ}{QP}$

Step 2: Identify Given Values

• $ST = 22$
• $TP = 55$
• $RQ = 39$

First, calculate $QP$: $QP = ST + TP = 22 + 55 = 77$

Step 3: Solve for $RS$

Using the proportion: $\frac{RS}{22} = \frac{39}{77}$

Solve for $RS$:

$RS = \frac{39}{77} \times 22$

$RS = \frac{39 \times 22}{77}$

$RS = \frac{858}{77} = 11.14$

Final Answer:

$RS \approx 11.14$

Would you like a detailed breakdown of any step?

Here are some related questions to explore further:

1. How does the Triangle Proportionality Theorem help in real-world applications?
2. What happens if the given lines are not parallel?
3. Can this theorem be extended to three-dimensional geometry?
4. How do you derive the Triangle Proportionality Theorem?
5. What other methods can be used to solve for $RS$?

Tip: Always check if the given lines are parallel before applying the Triangle Proportionality Theorem!