To find the coordinates of point [tex]\( R \)[/tex] which divides the line segment [tex]\(\overline{EF}\)[/tex] in the ratio [tex]\(1:5\)[/tex], we need to use the section formula. The section formula for a point dividing a line segment in the ratio [tex]\(m : n\)[/tex] is:
[tex]\[
(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\][/tex]
Given the coordinates of [tex]\(E\)[/tex] and [tex]\(F\)[/tex]:
[tex]\[ E = (4, 8) \][/tex]
[tex]\[ F = (11, 4) \][/tex]
The ratio [tex]\(1 : 5\)[/tex] corresponds to [tex]\(m = 1\)[/tex] and [tex]\(n = 5\)[/tex].
Substituting these values into the section formula:
[tex]\[
R_x = \frac{1 \cdot 11 + 5 \cdot 4}{1 + 5} = \frac{11 + 20}{6} = \frac{31}{6} \approx 5.17
\][/tex]
[tex]\[
R_y = \frac{1 \cdot 4 + 5 \cdot 8}{1 + 5} = \frac{4 + 40}{6} = \frac{44}{6} \approx 7.33
\][/tex]
Therefore, the coordinates of [tex]\(R\)[/tex] to two decimal places are:
[tex]\[
(5.17, 7.33)
\][/tex]
Hence, the correct answer is:
C. [tex]\((5.17, 7.33)\)[/tex]
