To find the coordinates of point [tex]\( E \)[/tex], which partitions the directed line segment from point [tex]\( J \)[/tex] to point [tex]\( K \)[/tex] into a ratio of [tex]\( 1 : 4 \)[/tex], we use the section formula in coordinate geometry.
Given:
- Coordinates of point [tex]\( J \)[/tex]: [tex]\( (x_1, y_1) = (-13, -3) \)[/tex]
- Coordinates of point [tex]\( K \)[/tex]: [tex]\( (x_2, y_2) = (17, 11) \)[/tex]
- Ratio [tex]\( J \)[/tex] to [tex]\( K \)[/tex]: [tex]\( 1 : 4 \)[/tex] (where [tex]\( m = 1 \)[/tex] and [tex]\( n = 4 \)[/tex])
The section formula states:
[tex]\[
x = \frac{m \cdot x_2 + n \cdot x_1}{m + n}
\][/tex]
[tex]\[
y = \frac{m \cdot y_2 + n \cdot y_1}{m + n}
\][/tex]
Substituting the given values into these formulas:
For the [tex]\( x \)[/tex]-coordinate of point [tex]\( E \)[/tex]:
[tex]\[
x = \frac{1 \cdot 17 + 4 \cdot (-13)}{1 + 4}
\][/tex]
Solving this:
[tex]\[
x = \frac{17 + (-52)}{5}
\][/tex]
[tex]\[
x = \frac{17 - 52}{5}
\][/tex]
[tex]\[
x = \frac{-35}{5}
\][/tex]
[tex]\[
x = -7
\][/tex]
For the [tex]\( y \)[/tex]-coordinate of point [tex]\( E \)[/tex]:
[tex]\[
y = \frac{1 \cdot 11 + 4 \cdot (-3)}{1 + 4}
\][/tex]
Solving this:
[tex]\[
y = \frac{11 + (-12)}{5}
\][/tex]
[tex]\[
y = \frac{11 - 12}{5}
\][/tex]
[tex]\[
y = \frac{-1}{5}
\][/tex]
[tex]\[
y = -0.2
\][/tex]
So, the coordinates of point [tex]\( E \)[/tex] are [tex]\( (-7, -0.2) \)[/tex].
