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], we can use the section formula. The section formula for a point dividing a line segment in a given ratio [tex]\( m:n \)[/tex] is:
[tex]\[
R\left( \frac{m \cdot F_x + n \cdot E_x}{m + n}, \frac{m \cdot F_y + n \cdot E_y}{m + n} \right)
\][/tex]
Here, the coordinates of points [tex]\( E \)[/tex] and [tex]\( F \)[/tex] are given as:
[tex]\[ E(x_1, y_1) = (4, 8) \][/tex]
[tex]\[ F(x_2, y_2) = (11, 4) \][/tex]
The ratio [tex]\( m:n \)[/tex] is given as [tex]\( 1:5 \)[/tex], meaning [tex]\( m = 1 \)[/tex] and [tex]\( n = 5 \)[/tex].
Let's calculate the coordinates of [tex]\( R \)[/tex]:
1. Calculate the [tex]\( x \)[/tex]-coordinate of [tex]\( R \)[/tex]:
[tex]\[
R_x = \frac{m \cdot F_x + n \cdot E_x}{m + n} = \frac{1 \cdot 11 + 5 \cdot 4}{1 + 5}
\][/tex]
Breaking it down step by step:
[tex]\[
R_x = \frac{11 + 20}{6} = \frac{31}{6} \approx 5.17
\][/tex]
2. Calculate the [tex]\( y \)[/tex]-coordinate of [tex]\( R \)[/tex]:
[tex]\[
R_y = \frac{m \cdot F_y + n \cdot E_y}{m + n} = \frac{1 \cdot 4 + 5 \cdot 8}{1 + 5}
\][/tex]
Breaking it down step by step:
[tex]\[
R_y = \frac{4 + 40}{6} = \frac{44}{6} \approx 7.33
\][/tex]
After performing these calculations, we find the coordinates of point [tex]\( R \)[/tex] to be approximately [tex]\( (5.17, 7.33) \)[/tex].
Thus, the correct answer is:
C. [tex]\( (5.17, 7.33) \)[/tex]
