Sure, let's find the [tex]\( y \)[/tex]-coordinate of the point that divides the directed line segment from [tex]\( J \)[/tex] to [tex]\( K \)[/tex] into a ratio of [tex]\( 5:1 \)[/tex]. We'll use the section formula.
The coordinates of point [tex]\( J \)[/tex] are [tex]\( (-8, 6) \)[/tex] and the coordinates of point [tex]\( K \)[/tex] are [tex]\( (0, -5) \)[/tex].
The section formula states that the coordinates of a point dividing a line segment internally in the ratio [tex]\( m:n \)[/tex] are given by:
[tex]\[
\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\][/tex]
In our case, the coordinates of [tex]\( J \)[/tex] are [tex]\( (x_1, y_1) = (-8, 6) \)[/tex] and the coordinates of [tex]\( K \)[/tex] are [tex]\( (x_2, y_2) = (0, -5) \)[/tex]. The ratio is [tex]\( m:n = 5:1 \)[/tex].
We need to find only the [tex]\( y \)[/tex]-coordinate using the formula:
[tex]\[
y = \left( \frac{m}{m+n} \right) (y_2 - y_1) + y_1.
\][/tex]
Plugging in the values:
[tex]\[
y = \left( \frac{5}{5+1} \right) (-5 - 6) + 6,
\][/tex]
[tex]\[
y = \left( \frac{5}{6} \right) (-11) + 6,
\][/tex]
[tex]\[
y = \left( \frac{5 \times -11}{6} \right) + 6,
\][/tex]
[tex]\[
y = \left( -\frac{55}{6} \right) + 6,
\][/tex]
[tex]\[
y = -9.166666666666668 + 6,
\][/tex]
[tex]\[
y = -3.166666666666668.
\][/tex]
So, the [tex]\( y \)[/tex]-coordinate of the point that divides the directed line segment from [tex]\( J \)[/tex] to [tex]\( K \)[/tex] in the ratio of [tex]\( 5:1 \)[/tex] is approximately [tex]\( -3.1667 \)[/tex].
