To find the [tex]\( y \)[/tex]-coordinate of the point that divides the directed line segment from [tex]\( J \)[/tex] to [tex]\( K \)[/tex] into a ratio of [tex]\( 5:1 \)[/tex], we can use the given formula:
[tex]\[ y = \left(\frac{m}{m+n}\right) \left(y_2 - y_1\right) + y_1 \][/tex]
Here, the values are given as follows:
- [tex]\( y_1 = -8 \)[/tex] (the [tex]\( y \)[/tex]-coordinate of point [tex]\( J \)[/tex])
- [tex]\( y_2 = 6 \)[/tex] (the [tex]\( y \)[/tex]-coordinate of point [tex]\( K \)[/tex])
- [tex]\( m = 5 \)[/tex] (the ratio of the segment closer to point [tex]\( K \)[/tex])
- [tex]\( n = 1 \)[/tex] (the ratio of the segment closer to point [tex]\( J \)[/tex])
Now, we will substitute these values into the formula and compute the results:
1. Calculate the term [tex]\( \left(\frac{m}{m+n}\right) \)[/tex]:
[tex]\[
\frac{m}{m+n} = \frac{5}{5+1} = \frac{5}{6}
\][/tex]
2. Compute the difference [tex]\( y_2 - y_1 \)[/tex]:
[tex]\[
y_2 - y_1 = 6 - (-8) = 6 + 8 = 14
\][/tex]
3. Multiply the fraction [tex]\( \left(\frac{m}{m+n}\right) \)[/tex] by the difference [tex]\( (y_2 - y_1) \)[/tex]:
[tex]\[
\left( \frac{5}{6} \right) \left( 14 \right) = \frac{5 \times 14}{6} = \frac{70}{6} \approx 11.666666666666668
\][/tex]
4. Add [tex]\( y_1 \)[/tex] to the product obtained in step 3:
[tex]\[
\left( \frac{5}{6} \left( y_2 - y_1 \right) + y_1 \right) = 11.666666666666668 - 8 = 3.666666666666668
\][/tex]
Thus, the [tex]\( y \)[/tex]-coordinate of the point that divides the directed line segment from [tex]\( J \)[/tex] to [tex]\( K \)[/tex] into a ratio of [tex]\( 5:1 \)[/tex] is:
[tex]\[ 3.666666666666668 \][/tex]
