Bot GPT 4-o mini In a triangle, the centroid (S) divides each median (the line segment from a vertex to the midpoint of the opposite side) in a ratio of 2:1. This means that the segment from the centroid to a vertex is twice the length of the segment from the centroid to the midpoint of the opposite side.
Here, we have:
- Length of segment \( NS = x + 10 \) feet (from vertex N to centroid S)
- Length of segment \( SR = x + 3 \) feet (from centroid S to the midpoint R of side NL)
Using the ratio in which the centroid divides the median, we can say:
\[ NS : SR = 2 : 1 \]
This implies:
\[ NS = 2 \times SR \]
Substituting the values in:
\[ x + 10 = 2(x + 3) \]
Now, simplify and solve for \( x \):
\[ x + 10 = 2x + 6 \]
Subtract \( x \) from both sides:
\[ 10 = x + 6 \]
Now, subtract 6 from both sides:
\[ x = 4 \]
Now, we can determine the lengths of \( NS \) and \( SR \):
\[ NS = x + 10 = 4 + 10 = 14 \text{ feet} \] \[ SR = x + 3 = 4 + 3 = 7 \text{ feet} \]
Now we need to find \( RS \). Since the centroid divides the median into two parts:
\[ RS = SR = 7 \text{ feet} \]
So, the length of segment \( RS \) is 7 feet.
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