To determine the coordinates of the point [tex]\( R \)[/tex] which divides the line segment [tex]\( \overline{EF} \)[/tex] in the ratio [tex]\( 1:5 \)[/tex], we will use the section formula. The section formula in a two-dimensional coordinate system is used to find the coordinates of a point dividing a line segment joining two given points in a given ratio.
Given the points:
- [tex]\( E = (4, 8) \)[/tex]
- [tex]\( F = (11, 4) \)[/tex]
And the ratio [tex]\( m:n = 1:5 \)[/tex].
The section formula for a point [tex]\( R = (R_x, R_y) \)[/tex] dividing the line segment joining points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the ratio [tex]\( m:n \)[/tex] internally is given by:
[tex]\[ R_x = \frac{mx_2 + nx_1}{m+n} \][/tex]
[tex]\[ R_y = \frac{my_2 + ny_1}{m+n} \][/tex]
Substitute the given values into the formulas:
1. Calculate the [tex]\( x \)[/tex]-coordinate [tex]\( R_x \)[/tex]:
- [tex]\( x_1 = 4 \)[/tex]
- [tex]\( x_2 = 11 \)[/tex]
- [tex]\( m = 1 \)[/tex]
- [tex]\( n = 5 \)[/tex]
[tex]\[ R_x = \frac{(1 \cdot 11) + (5 \cdot 4)}{1 + 5} \][/tex]
Simplify the expression:
[tex]\[ R_x = \frac{11 + 20}{6} \][/tex]
[tex]\[ R_x = \frac{31}{6} \][/tex]
[tex]\[ R_x \approx 5.17 \][/tex]
2. Calculate the [tex]\( y \)[/tex]-coordinate [tex]\( R_y \)[/tex]:
- [tex]\( y_1 = 8 \)[/tex]
- [tex]\( y_2 = 4 \)[/tex]
- [tex]\( m = 1 \)[/tex]
- [tex]\( n = 5 \)[/tex]
[tex]\[ R_y = \frac{(1 \cdot 4) + (5 \cdot 8)}{1 + 5} \][/tex]
Simplify the expression:
[tex]\[ R_y = \frac{4 + 40}{6} \][/tex]
[tex]\[ R_y = \frac{44}{6} \][/tex]
[tex]\[ R_y \approx 7.33 \][/tex]
Therefore, the coordinates of point [tex]\( R \)[/tex] are approximately [tex]\( (5.17, 7.33) \)[/tex].
The correct answer, matching these coordinates, is:
C. [tex]\( (5.17, 7.33) \)[/tex]
