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 of [tex]$5:1$[/tex], we will use the section formula.
The section formula states that if a point divides the line segment joining [tex]$(x_1, y_1)$[/tex] and [tex]$(x_2, y_2)$[/tex] in the ratio [tex]$m:n$[/tex], then the coordinates of the dividing point are:
[tex]$
\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)
$[/tex]
Here, we are only interested in the [tex]$y$[/tex]-coordinate, given by:
[tex]$
y = \left(\frac{m}{m+n}\right)(y_2 - y_1) + y_1
$[/tex]
Given the following information:
- The ratio [tex]$m:n = 5:1$[/tex],
- [tex]$y_1 = -8$[/tex],
- [tex]$y_2 = 0$[/tex],
We will substitute these values into the formula:
[tex]$
y = \left(\frac{5}{5+1}\right)(0 - (-8)) + (-8)
$[/tex]
Now, simplifying step-by-step:
1. Calculate the denominator of the fraction: [tex]$5 + 1 = 6$[/tex].
2. Should substitute the values:
[tex]$
y = \left(\frac{5}{6}\right)(0 + 8) - 8
$[/tex]
3. Simplify further:
[tex]$
y = \left(\frac{5}{6}\right)(8) - 8
$[/tex]
4. Calculate [tex]$\left(\frac{5}{6}\right) \times 8$[/tex]:
[tex]$
y = \frac{40}{6} - 8 = \frac{40}{6} - \frac{48}{6}
$[/tex]
5. Subtract the fractions:
[tex]$
y = \frac{40 - 48}{6} = \frac{-8}{6} = -\frac{4}{3} \approx -1.3333
$[/tex]
Therefore, the [tex]$y$[/tex]-coordinate of the point that divides the directed line segment from [tex]$J$[/tex] to [tex]$K$[/tex] into the ratio of [tex]$5:1$[/tex] is:
[tex]$
y = -1.3333
$[/tex]
