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 can use the section formula. Here is a step-by-step process:
1. Identify the given information:
- The ratio [tex]\( m:n \)[/tex] is [tex]\( 5:1 \)[/tex].
- Let the [tex]\( y \)[/tex]-coordinates of points [tex]\( J \)[/tex] and [tex]\( K \)[/tex] be [tex]\( y_1 \)[/tex] and [tex]\( y_2 \)[/tex] respectively.
- From the question, we know some possible values for [tex]\( y_1 \)[/tex] and [tex]\( y_2 \)[/tex]: [tex]\( -8 \)[/tex], [tex]\( -5 \)[/tex], [tex]\( 0 \)[/tex], [tex]\( 6 \)[/tex].
2. Use the section formula for [tex]\( y \)[/tex]-coordinate:
The formula for the [tex]\( y \)[/tex]-coordinate of the point that divides the segment in the ratio [tex]\( m:n \)[/tex] is:
[tex]\[
y = \frac{m \cdot y_2 + n \cdot y_1}{m + n}
\][/tex]
For this specific problem, we are given the ratio [tex]\( 5:1 \)[/tex], so [tex]\( m = 5 \)[/tex] and [tex]\( n = 1 \)[/tex].
3. Select specific [tex]\( y_1 \)[/tex] and [tex]\( y_2 \)[/tex]:
Let's use [tex]\( y_1 = -8 \)[/tex] and [tex]\( y_2 = 6 \)[/tex].
4. Substitute the given values into the formula:
[tex]\[
y = \frac{(5 \cdot 6) + (1 \cdot -8)}{5 + 1} = \frac{30 + (-8)}{6} = \frac{22}{6} = 3.6666666666666665
\][/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]\( 3.6666666666666665 \)[/tex].
