Sure, let's solve this problem step-by-step.
Step 1: Understanding the Ratio
We are given a directed line segment from point [tex]\( J \)[/tex] to point [tex]\( K \)[/tex]. The segment is divided in the ratio [tex]\( 5 : 1 \)[/tex]. This means for every 5 parts along the segment, there is 1 part to complete the segment.
Step 2: Identifying the Coordinates
We are given that:
- The starting [tex]\( y \)[/tex]-coordinate ([tex]\( y_1 \)[/tex]) is -8.
- The ending [tex]\( y \)[/tex]-coordinate ([tex]\( y_2 \)[/tex]) is 6.
Step 3: Applying the Formula
The formula given is:
[tex]\[
v = \left(\frac{m}{m+n}\right)(v_2 - v_1) + v_1
\][/tex]
where:
- [tex]\( m \)[/tex] is the first part of the ratio (5),
- [tex]\( n \)[/tex] is the second part of the ratio (1),
- [tex]\( v_1 \)[/tex] and [tex]\( v_2 \)[/tex] are the [tex]\( y \)[/tex]-coordinates of point [tex]\( J \)[/tex] and [tex]\( K \)[/tex] respectively.
Step 4: Performing Substitution
Substitute the values into the formula:
[tex]\[
y = \left( \frac{5}{5 + 1} \right) ( 6 - (-8) ) + (-8)
\][/tex]
Simplify the fraction:
[tex]\[
y = \left( \frac{5}{6} \right) ( 6 + 8 ) - 8
\][/tex]
Simplify inside the parenthesis:
[tex]\[
y = \left( \frac{5}{6} \right) ( 14 ) - 8
\][/tex]
Step 5: Multiplication
Multiply [tex]\( \frac{5}{6} \)[/tex] by 14:
[tex]\[
y = \frac{5 \times 14}{6} - 8
\][/tex]
[tex]\[
y = \frac{70}{6} - 8
\][/tex]
Step 6: Simplifying the Fraction
[tex]\[
\frac{70}{6} \approx 11.666666666666668
\][/tex]
Step 7: Final Calculation
Subtract 8:
[tex]\[
y = 11.666666666666668 - 8
\][/tex]
[tex]\[
y \approx 3.666666666666668
\][/tex]
So, 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 approximately [tex]\( 3.666666666666668 \)[/tex].
This value matches with the provided result. Therefore, the final answer is approximately [tex]\( 3.67 \)[/tex].
Let me know if I can help you further!
