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 the detailed solution step-by-step:
Given coordinates:
- Let [tex]\( v_1 = -8 \)[/tex] (the [tex]\( y \)[/tex]-coordinate of point [tex]\( J \)[/tex])
- Let [tex]\( v_2 = 0 \)[/tex] (the [tex]\( y \)[/tex]-coordinate of point [tex]\( K \)[/tex])
We need to find the [tex]\( y \)[/tex]-coordinate of the point that divides the segment in the ratio [tex]\( 5:1 \)[/tex].
The section formula for finding a point that divides a 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]
Let's use the given information to plug into the formula:
1. [tex]\( m = 5 \)[/tex]
2. [tex]\( n = 1 \)[/tex]
3. [tex]\( v_1 = -8 \)[/tex]
4. [tex]\( v_2 = 0 \)[/tex]
Now, we substitute these values into the formula:
[tex]\[
v = \left( \frac{5}{5+1} \right) \left( 0 - (-8) \right) + (-8)
\][/tex]
Simplify inside the parentheses first:
[tex]\[
0 - (-8) = 0 + 8 = 8
\][/tex]
Next, perform the multiplication and addition:
[tex]\[
v = \left( \frac{5}{6} \right) (8) + (-8)
\][/tex]
[tex]\[
v = \frac{5 \times 8}{6} + (-8)
\][/tex]
[tex]\[
v = \frac{40}{6} - 8
\][/tex]
Convert the fraction to a decimal:
[tex]\[
\frac{40}{6} \approx 6.66666666667
\][/tex]
Now, subtract 8 from approximately [tex]\( 6.66666666667 \)[/tex]:
[tex]\[
v \approx 6.66666666667 - 8
\][/tex]
[tex]\[
v \approx -1.33333333333
\][/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 a ratio [tex]\( 5:1 \)[/tex] is approximately:
[tex]\[
-1.33333333333
\][/tex]
