In triangle NQL, point S is the centroid, NS = (x + 10) feet, and SR = (x + 3) feet.

Question

Triangle N Q L has centroid S. Lines are drawn from each point to the midpoint of the opposite side to form line segments N R, Q M, and L P. The length of line segment N S is x + 10 and the length of line segment S R is x + 3.

What is RS?

4 feet
7 feet
10 feet
14 feet

Answer 1

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.