To find the coordinates of a point [tex]\( R \)[/tex] that divides a line segment [tex]\( \overline{EF} \)[/tex] in a given ratio, we use the section formula. The section formula states that if a point [tex]\( R(x, y) \)[/tex] divides a line segment joining [tex]\( E(x_1, y_1) \)[/tex] and [tex]\( F(x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex], then the coordinates of [tex]\( R \)[/tex] are given by:
[tex]\[
R_x = \frac{m x_2 + n x_1}{m + n}
\][/tex]
[tex]\[
R_y = \frac{m y_2 + n y_1}{m + n}
\][/tex]
Given:
- Coordinates of [tex]\( E \)[/tex]: [tex]\( E(4, 8) \)[/tex]
- Coordinates of [tex]\( F \)[/tex]: [tex]\( F(11, 4) \)[/tex]
- Ratio [tex]\( m : n = 1 : 5 \)[/tex]
Let's plug in these values into the formulas.
### Calculation for [tex]\( R_x \)[/tex]:
\begin{aligned}
R_x & = \frac{1 \times 11 + 5 \times 4}{1 + 5} \\
& = \frac{11 + 20}{6} \\
& = \frac{31}{6} \\
& = 5.17 \\
\end{aligned}
### Calculation for [tex]\( R_y \)[/tex]:
\begin{aligned}
R_y & = \frac{1 \times 4 + 5 \times 8}{1 + 5} \\
& = \frac{4 + 40}{6} \\
& = \frac{44}{6} \\
& = 7.33 \\
\end{aligned}
Hence, the coordinates of point [tex]\( R \)[/tex] are [tex]\((5.17, 7.33)\)[/tex].
So the correct answer is:
C. [tex]\((5.17, 7.33)\)[/tex]
