To find the coordinates of point [tex]\( R \)[/tex] that divides the line segment [tex]\( \overline{EF} \)[/tex] in the ratio 1:5, where [tex]\( E \)[/tex] and [tex]\( F \)[/tex] have coordinates [tex]\((4,8)\)[/tex] and [tex]\((11,4)\)[/tex] respectively, we use the section formula for internal division.
The section formula 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, [tex]\( (x_1, y_1) \)[/tex] are the coordinates of [tex]\( E \)[/tex] which are [tex]\((4,8)\)[/tex], and [tex]\( (x_2, y_2) \)[/tex] are the coordinates of [tex]\( F \)[/tex] which are [tex]\((11,4)\)[/tex]. The ratio [tex]\( m:n = 1:5 \)[/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].
So, the correct answer is:
C. [tex]\( (5.17, 7.33) \)[/tex]
