To find the point that partitions the directed line segment [tex]\( \overline{AB} \)[/tex] into a [tex]\( 2:3 \)[/tex] ratio, we will employ the section formula for internal division. This formula provides a way to determine the coordinates of a point dividing a line segment internally in a given ratio.
Given the points:
- [tex]\( A = (2, 6) \)[/tex]
- [tex]\( B = (18, 12) \)[/tex]
The ratio [tex]\( m:n \)[/tex] is given as [tex]\( 2:3 \)[/tex].
The coordinates [tex]\((x, y)\)[/tex] of the point that divides the line segment [tex]\( \overline{AB} \)[/tex] in the ratio [tex]\( m:n \)[/tex] are given by:
[tex]\[
\left( \frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n} \right)
\][/tex]
Plugging in the given values:
1. For the [tex]\( x \)[/tex]-coordinate:
[tex]\[
x = \frac{2 \cdot 18 + 3 \cdot 2}{2 + 3} = \frac{36 + 6}{5} = \frac{42}{5} = 8.4
\][/tex]
2. For the [tex]\( y \)[/tex]-coordinate:
[tex]\[
y = \frac{2 \cdot 12 + 3 \cdot 6}{2 + 3} = \frac{24 + 18}{5} = \frac{42}{5} = 8.4
\][/tex]
So, the coordinates of the point that partitions the line segment [tex]\( \overline{AB} \)[/tex] in a [tex]\( 2:3 \)[/tex] ratio are [tex]\( (8.4, 8.4) \)[/tex].
Now we need to match these coordinates with the given options:
1. [tex]\(\left(8 \frac{2}{5}, 8 \frac{2}{5}\right) \Rightarrow (8 + 0.4, 8 + 0.4) = (8.4, 8.4)\)[/tex]
2. [tex]\(\left(10 \frac{1}{2}, 10 \frac{1}{2}\right) \Rightarrow (10.5, 10.5)\)[/tex]
3. [tex]\(\left(11 \frac{3}{9}, 9 \frac{3}{5}\right) \Rightarrow (11.333, 9.6)\)[/tex]
4. [tex]\(\left(14 \frac{1}{2}, 12\right) \Rightarrow (14.5, 12)\)[/tex]
The option matching [tex]\( (8.4, 8.4) \)[/tex] is [tex]\(\left(8 \frac{2}{5}, 8 \frac{2}{5}\right)\)[/tex].
Hence, the point that partitions the directed line segment [tex]\( \overline{AB} \)[/tex] into a [tex]\( 2:3 \)[/tex] ratio is:
[tex]\[
\boxed{\left(8 \frac{2}{5}, 8 \frac{2}{5}\right)}
\][/tex]
