The problem states that $Z$ is the centroid of triangle $TRS$. Given that $TO = 7b + 5$, $OR = 13b - 10$, and $TR = 18b$, we need to find the value of $b$ and the length of $TR$. Since $Z$ is the centroid, $O$ is the midpoint of side $TR$, which means $TO = OR$.

1. Problem Description

The problem states that $Z$ is the centroid of triangle $TRS$. Given that $TO = 7b + 5$, $OR = 13b - 10$, and $TR = 18b$, we need to find the value of $b$ and the length of $TR$. Since $Z$ is the centroid, $O$ is the midpoint of side $TR$, which means $TO = OR$.

2. Solution Steps

Since $O$ is the midpoint of $TR$, $TO = OR$. We can set up the equation:

$7b + 5 = 13b - 10$

Subtract $7b$ from both sides:

$5 = 6b - 10$

Add $10$ to both sides:

$15 = 6b$

Divide both sides by $6$:

$b = \frac{15}{6} = \frac{5}{2} = 2.5$

Now we can find the length of $TR$:

$TR = 18b = 18 \times \frac{5}{2} = 9 \times 5 = 45$

3. Final Answer

$b = 2.5$

$TR = 45$