To find the coordinates of point \( S \) that divides the line segment \( QR \) in the ratio \( QS:RS = 3:1 \), we can use the section formula.
The section formula states that if a point \( S \) divides the line segment joining points \( Q(x_1, y_1) \) and \( R(x_2, y_2) \) in the ratio \( m:n \), the coordinates of \( S \) can be found using:
\[ S\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
Here, \( Q(6, 2) \) and \( R(18, 18) \). The ratio \( QS:RS = 3:1 \) means \( m = 3 \) and \( n = 1 \).
Substituting the values into the formula:
-
For the x-coordinate: \[ x_S = \frac{3 \cdot 18 + 1 \cdot 6}{3 + 1} = \frac{54 + 6}{4} = \frac{60}{4} = 15 \]
-
For the y-coordinate: \[ y_S = \frac{3 \cdot 18 + 1 \cdot 2}{3 + 1} = \frac{54 + 2}{4} = \frac{56}{4} = 14 \]
Thus, the coordinates of point \( S \) are:
\[ \boxed{(15, 14)} \]