In the given diagram, we have a triangle $\triangle PQR$ with points $S$ and $T$ on the sides. It looks like we need to solve for $x$ using the information provided.

The side lengths and line segments are labeled as follows:

• $PS = 36$
• $ST = 30$
• $PR = 5x + 13$
• $QR = 6x - 2$

This configuration resembles a case where the triangle proportionality theorem might apply, which states that if a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally. Here, $ST$ is likely parallel to $QR$, which allows us to set up proportional relationships.

We have:

$\frac{PS}{PR} = \frac{ST}{QR}$

Substituting the given lengths into this equation:

$\frac{36}{5x + 13} = \frac{30}{6x - 2}$

Now, I'll solve this equation for $x$.The value of $x$ is 7.

Would you like further details on this solution or have any other questions?

Here are 5 related questions you might find interesting:

1. How does the triangle proportionality theorem apply in other triangle configurations?
2. What happens if $ST$ is not parallel to $QR$?
3. Can we verify the solution by substituting $x = 7$ back into the original proportions?
4. What are some common theorems related to similar triangles?
5. How can we use proportionality to solve problems in other geometric shapes?

Tip: Proportionality is a powerful tool in geometry, especially when dealing with triangles. Look for parallel lines and corresponding ratios to simplify complex problems.