To find 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 [tex]\( 5:1 \)[/tex], we use the section formula for the coordinates of a point dividing a line segment internally.
Let [tex]\( J = (0, -8) \)[/tex] and [tex]\( K = (1, 6) \)[/tex]. We are given the ratio [tex]\( 5:1 \)[/tex], where [tex]\( m = 5 \)[/tex] and [tex]\( n = 1 \)[/tex].
The section formula for the [tex]\( y \)[/tex]-coordinate is given by:
[tex]\[
y = \left(\frac{m}{m+n}\right)(y_2 - y_1) + y_1
\][/tex]
Plugging in the coordinates and the ratio, we have:
[tex]\[
y_1 = -8, \quad y_2 = 6
\][/tex]
First, compute the fraction of the ratio:
[tex]\[
\frac{m}{m+n} = \frac{5}{5+1} = \frac{5}{6}
\][/tex]
Next, calculate the difference [tex]\( y_2 - y_1 \)[/tex]:
[tex]\[
y_2 - y_1 = 6 - (-8) = 6 + 8 = 14
\][/tex]
Now, apply these values into the section formula:
[tex]\[
y = \left(\frac{5}{6}\right) \cdot 14 + (-8)
\][/tex]
Compute the multiplication:
[tex]\[
\left(\frac{5}{6}\right) \cdot 14 = \frac{70}{6} \approx 11.666666666666668
\][/tex]
Finally, add this result to [tex]\( y_1 \)[/tex]:
[tex]\[
y = 11.666666666666668 - 8 = 3.666666666666668
\][/tex]
Therefore, 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 [tex]\( 5:1 \)[/tex] is:
[tex]\[
3.666666666666668
\][/tex]
