To find the [tex]$y$[/tex]-coordinate of the point that divides the directed line segment from point [tex]\( J \)[/tex] to point [tex]\( K \)[/tex] in the ratio [tex]\( 5:1 \)[/tex], we use the formula:
[tex]\[ v = \left( \frac{m}{m+n} \right) (v_2 - v_1) + v_1 \][/tex]
Let's denote:
- [tex]\( v_1 \)[/tex] as the [tex]$y$[/tex]-coordinate of point [tex]\( J \)[/tex]
- [tex]\( v_2 \)[/tex] as the [tex]$y$[/tex]-coordinate of point [tex]\( K \)[/tex]
- [tex]\( m \)[/tex] and [tex]\( n \)[/tex] as the parts of the ratio
Given:
- Coordinates of point [tex]\( J \)[/tex]: [tex]\( (x_1, 0) \)[/tex] so [tex]\( v_1 = 0 \)[/tex]
- Coordinates of point [tex]\( K \)[/tex]: [tex]\( (x_2, -6) \)[/tex] so [tex]\( v_2 = -6 \)[/tex]
- The ratio [tex]\( m:n = 5:1 \)[/tex] so [tex]\( m = 5 \)[/tex] and [tex]\( n = 1 \)[/tex]
Substitute these values into the formula:
[tex]\[ v = \left( \frac{5}{5+1} \right) (-6 - 0) + 0 \][/tex]
Now calculate the coefficients and the expression inside the parentheses:
[tex]\[ v = \left( \frac{5}{6} \right) (-6) \][/tex]
Next, perform the multiplication:
[tex]\[ v = \frac{5}{6} \times (-6) \][/tex]
Simplify the expression:
[tex]\[ v = -5 \][/tex]
Therefore, the [tex]$y$[/tex]-coordinate of the point that divides the line segment from [tex]\( J \)[/tex] to [tex]\( K \)[/tex] in the ratio of [tex]\( 5:1 \)[/tex] is:
[tex]\[ \boxed{-5} \][/tex]
