Triangle N Q L has centroid S. Lines are drawn from each point through the centroid to the midpoint of the opposite side to form line segments N R, L P, and Q M. The length of line segment N S is 9 n minus 12 and the length of line segment S R is 3 n.
S is the centroid of triangle NQL.

What is the length of NR?

4 units
8 units
36 units
48 units

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.

Triangle N Q L has centroid S. Lines are drawn from each point through the centroid to the midpoint of the opposite side to form line segments N R, L P, and Q M. The length of line segment N S is 9n - 12 and the length of line segment S R is 3n. S is the centroid of triangle NQL. What is the length of NR? 4 units 8 units 36 units 48 units

Learn how to find the length of line segment NR in a triangle where the centroid divides the medians into a 2:1 ratio. This problem uses the concepts of geometry and centroid theorems. Perfect for students in Grades 9-11, this step-by-step solution explains how to apply the 2:1 ratio rule and calculate the exact length of NR.

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