To find 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 helps to find the coordinates of a point dividing a line segment in a given ratio.
Given:
- Coordinates of [tex]\( E \)[/tex] are [tex]\((4, 8)\)[/tex]
- Coordinates of [tex]\( F \)[/tex] are [tex]\((11, 4)\)[/tex]
- The ratio in which [tex]\(R\)[/tex] divides [tex]\(\overline{EF}\)[/tex] is [tex]\(1:5\)[/tex]
The section formula for a point [tex]\( (x, y) \)[/tex] dividing a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the ratio [tex]\(m:n\)[/tex] is given by:
[tex]\[
(x, y) = \left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right)
\][/tex]
Here, [tex]\( (x_1, y_1) = (4, 8) \)[/tex], [tex]\( (x_2, y_2) = (11, 4) \)[/tex], [tex]\( m = 1 \)[/tex], and [tex]\( n = 5 \)[/tex].
Let's calculate the coordinates step by step.
#### Step 1: Calculate [tex]\( x \)[/tex]-coordinate
[tex]\[
x_R = \frac{m \cdot x_2 + n \cdot x_1}{m + n} = \frac{1 \cdot 11 + 5 \cdot 4}{1 + 5} = \frac{11 + 20}{6} = \frac{31}{6} \approx 5.17
\][/tex]
#### Step 2: Calculate [tex]\( y \)[/tex]-coordinate
[tex]\[
y_R = \frac{m \cdot y_2 + n \cdot y_1}{m + n} = \frac{1 \cdot 4 + 5 \cdot 8}{1 + 5} = \frac{4 + 40}{6} = \frac{44}{6} \approx 7.33
\][/tex]
So, the coordinates of [tex]\( R \)[/tex] are [tex]\((5.17, 7.33)\)[/tex].
None of these are approximately equal to [tex]\((9.65, 4.77)\)[/tex]
