Sure, let's solve this step-by-step.
We are given:
- The coordinates of point J are [tex]\((-8, y_1)\)[/tex], let's denote [tex]\(y_1 = y_J\)[/tex].
- The coordinates of point K are [tex]\((y_2, 6)\)[/tex], let's denote [tex]\(y_2 = y_K\)[/tex].
- We want to find the y-coordinate of the point that divides the segment from J to K in the ratio [tex]\(5:1\)[/tex].
From the problem, the formula to find this coordinate is:
[tex]\[ v = \left(\frac{m}{m+n}\right)(v_2 - v_1) + v_1 \][/tex]
Here:
- [tex]\( m = 5 \)[/tex]
- [tex]\( n = 1 \)[/tex]
- [tex]\( y_1 = -8 \)[/tex]
- [tex]\( y_2 = 6 \)[/tex]
Now, we'll substitute these values into the formula:
[tex]\[ y = \left(\frac{m}{m+n}\right)(y_K - y_J) + y_J \][/tex]
Substitute [tex]\(m, n, y_J,\)[/tex] and [tex]\(y_K\)[/tex]:
[tex]\[ y = \left(\frac{5}{5+1}\right)(6 - (-8)) + (-8) \][/tex]
[tex]\[ y = \left(\frac{5}{6}\right)(6 + 8) - 8 \][/tex]
[tex]\[ y = \left(\frac{5}{6}\right)(14) - 8 \][/tex]
[tex]\[ y = \left(\frac{70}{6}\right) - 8 \][/tex]
[tex]\[ y = 11.666666666666668 - 8 \][/tex]
[tex]\[ y = 3.666666666666668 \][/tex]
Hence, the y-coordinate of the point that divides the directed line segment from J to K into a ratio of [tex]\(5:1\)[/tex] is approximately [tex]\(3.67\)[/tex] (or precisely [tex]\(3.666666666666668\)[/tex]).
