Certainly! To determine the coordinates of point [tex]\( R \)[/tex] that divides the segment [tex]\( \overline{EF} \)[/tex] in the ratio [tex]\( 1:5 \)[/tex], we can use the section formula. The section formula allows us to find a point that divides a line segment between two given points in a specified ratio.
The section formula in the case where point [tex]\( R \)[/tex] divides segment [tex]\( \overline{EF} \)[/tex] in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[
R_x = \frac{m \cdot F_x + n \cdot E_x}{m + n}
\][/tex]
[tex]\[
R_y = \frac{m \cdot F_y + n \cdot E_y}{m + n}
\][/tex]
Here, the coordinates of [tex]\( E \)[/tex] are [tex]\( (4, 8) \)[/tex], and the coordinates of [tex]\( F \)[/tex] are [tex]\( (11, 4) \)[/tex]. The ratio [tex]\( m:n \)[/tex] is [tex]\( 1:5 \)[/tex].
Now, let's calculate the coordinates of [tex]\( R \)[/tex].
First, we find the [tex]\( x \)[/tex]-coordinate of [tex]\( R \)[/tex]:
[tex]\[
R_x = \frac{1 \cdot 11 + 5 \cdot 4}{1 + 5}
\][/tex]
[tex]\[
R_x = \frac{11 + 20}{6}
\][/tex]
[tex]\[
R_x = \frac{31}{6}
\][/tex]
[tex]\[
R_x = 5.17 \text{ (rounded to two decimal places)}
\][/tex]
Next, we find the [tex]\( y \)[/tex]-coordinate of [tex]\( R \)[/tex]:
[tex]\[
R_y = \frac{1 \cdot 4 + 5 \cdot 8}{1 + 5}
\][/tex]
[tex]\[
R_y = \frac{4 + 40}{6}
\][/tex]
[tex]\[
R_y = \frac{44}{6}
\][/tex]
[tex]\[
R_y = 7.33 \text{ (rounded to two decimal places)}
\][/tex]
So, the coordinates of point [tex]\( R \)[/tex] are [tex]\( (5.17, 7.33) \)[/tex].
Therefore, the correct answer is:
C. [tex]\( (5.17, 7.33) \)[/tex]
