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]$5:11$[/tex], we'll use the section formula. Here is a step-by-step solution:
1. Identify the coordinates and the ratio:
- For point [tex]$J$[/tex], the [tex]$y$[/tex]-coordinate is [tex]\( y_J = -8 \)[/tex].
- For point [tex]$K$[/tex], the [tex]$y$[/tex]-coordinate is [tex]\( y_K = 6 \)[/tex].
- The given ratio is [tex]\( 5:11 \)[/tex], meaning [tex]\( m = 5 \)[/tex] and [tex]\( n = 11 \)[/tex].
2. Apply the section formula:
The section formula for finding the coordinate of a point dividing the line segment in the ratio [tex]\( m:n \)[/tex] is:
[tex]\[
v = \left( \frac{m}{m+n} \right) \left( v_2 - v_1 \right) + v_1
\][/tex]
Here, [tex]\( v_1 = y_J \)[/tex] and [tex]\( v_2 = y_K \)[/tex].
3. Substitute the values into the formula:
[tex]\[
y = \left( \frac{5}{5+11} \right) (6 - (-8)) + (-8)
\][/tex]
4. Simplify the expression:
- Calculate the denominator: [tex]\( 5 + 11 = 16 \)[/tex].
- Calculate the numerator inside the parentheses: [tex]\( 6 - (-8) = 6 + 8 = 14 \)[/tex].
5. Substitute back into the formula:
[tex]\[
y = \left( \frac{5}{16} \right) \cdot 14 + (-8)
\][/tex]
6. Perform the multiplication:
[tex]\[
y = \left( \frac{5 \cdot 14}{16} \right) + (-8) = \frac{70}{16} - 8
\][/tex]
7. Simplify the fraction:
[tex]\[
y = \frac{70}{16} = 4.375
\][/tex]
8. Complete the calculation:
[tex]\[
y = 4.375 - 8 = -3.625
\][/tex]
Thus, the [tex]\( y \)[/tex]-coordinate of the point dividing the directed line segment from [tex]\( J \)[/tex] to [tex]\( K \)[/tex] in the ratio [tex]\( 5:11 \)[/tex] is [tex]\(-3.625\)[/tex].
