To determine the coordinates of point [tex]\( R \)[/tex] that divides the line segment [tex]\( \overline{EF} \)[/tex] in the ratio [tex]\( 1:5 \)[/tex], we use the section formula. The section formula for dividing a line segment in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[
(x_R, y_R) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\][/tex]
Here, point [tex]\( E \)[/tex] has coordinates [tex]\( (x_1, y_1) = (4, 8) \)[/tex] and point [tex]\( F \)[/tex] has coordinates [tex]\( (x_2, y_2) = (11, 4) \)[/tex]. The ratio [tex]\( 1:5 \)[/tex] means [tex]\( m=1 \)[/tex] and [tex]\( n=5 \)[/tex].
Substituting these values into the section formula:
[tex]\[
x_R = \frac{1 \cdot 11 + 5 \cdot 4}{1+5} = \frac{11 + 20}{6} = \frac{31}{6} \approx 5.17
\][/tex]
[tex]\[
y_R = \frac{1 \cdot 4 + 5 \cdot 8}{1+5} = \frac{4 + 40}{6} = \frac{44}{6} \approx 7.33
\][/tex]
Thus, the coordinates of point [tex]\( R \)[/tex] are approximately [tex]\((5.17, 7.33)\)[/tex].
The correct answer is:
C. [tex]\((5.17, 7.33)\)[/tex]
