In a triangle, the centroid $S$ divides each median into a ratio of 2:1. This means that the segment from a vertex to the centroid is twice the length of the segment from the centroid to the midpoint of the opposite side.

Given:

• $N$ is a vertex.
• $R$ is the midpoint of the side opposite $N$.
• $NS$ (the segment from $N$ to the centroid $S$) is given as $9n - 12$.
• $SR$ (the segment from $S$ to $R$) is given as $3n$.

Since the centroid divides the median into a 2:1 ratio: $\frac{NS}{SR} = \frac{2}{1}$

We can use this relationship to set up the equation: $\frac{9n - 12}{3n} = \frac{2}{1}$

Cross-multiplying gives: $9n - 12 = 6n$

Subtract $6n$ from both sides: $3n - 12 = 0$

Add 12 to both sides: $3n = 12$

Divide by 3: $n = 4$

Now that we have $n = 4$, we can find the length of $NS$ and $SR$: $NS = 9n - 12 = 9(4) - 12 = 36 - 12 = 24$

$SR = 3n = 3 \times 4 = 12$

The entire length of $NR$ is the sum of $NS$ and $SR$: $NR = NS + SR = 24 + 12 = 36$

So, the length of $NR$ is 36 units.

Answer: 36 units

Would you like more details or have any questions? Here are five related questions to explore further:

1. How does the centroid divide each median in a triangle?
2. What is the formula for finding the centroid of a triangle given the coordinates of its vertices?
3. How does the centroid affect the balance point of a triangle?
4. Can a triangle have more than one centroid? Why or why not?
5. What are the properties of a median in a triangle?

Tip: Remember that the centroid of a triangle always divides each median into a ratio of 2:1, with the longer segment always being closer to the vertex.