To determine the coordinates of point [tex]\( R \)[/tex] which divides the line segment [tex]\(\overline{EF}\)[/tex] in the ratio [tex]\(1:5\)[/tex], given [tex]\( E = (4, 8) \)[/tex] and [tex]\( F = (11, 4) \)[/tex], we can use the section formula.
The section formula for a point [tex]\( R = (x, y) \)[/tex] dividing a line segment [tex]\(\overline{EF}\)[/tex] in the ratio [tex]\( \frac{m}{n} \)[/tex] is given by:
[tex]\[
R = \left( \frac{m \cdot x_2 + n \cdot x_1}{m+n}, \frac{m \cdot y_2 + n \cdot y_1}{m+n} \right)
\][/tex]
Here, [tex]\( (x_1, y_1) \)[/tex] are the coordinates of [tex]\( E \)[/tex], [tex]\( (x_2, y_2) \)[/tex] are the coordinates of [tex]\( F \)[/tex], [tex]\( m = 1 \)[/tex], and [tex]\( n = 5 \)[/tex].
Substituting the given points and the ratio into the formula:
[tex]\[
x_R = \frac{1 \cdot 11 + 5 \cdot 4}{1 + 5}
\][/tex]
[tex]\[
x_R = \frac{11 + 20}{6}
\][/tex]
[tex]\[
x_R = \frac{31}{6}
\][/tex]
[tex]\[
x_R \approx 5.17
\][/tex]
Similarly,
[tex]\[
y_R = \frac{1 \cdot 4 + 5 \cdot 8}{1 + 5}
\][/tex]
[tex]\[
y_R = \frac{4 + 40}{6}
\][/tex]
[tex]\[
y_R = \frac{44}{6}
\][/tex]
[tex]\[
y_R \approx 7.33
\][/tex]
Therefore, the coordinates of point [tex]\( R \)[/tex] to two decimal places are [tex]\((5.17, 7.33)\)[/tex].
The correct answer is:
C. [tex]\((5.17, 7.33)\)[/tex]
