What is 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]?

A. [tex]\(-8\)[/tex]
B. [tex]\(-5\)[/tex]
C. 0
D. 6



Answer :

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 use the section formula for the coordinates of a point dividing a line segment internally.

Let [tex]\( J = (0, -8) \)[/tex] and [tex]\( K = (1, 6) \)[/tex]. We are given the ratio [tex]\( 5:1 \)[/tex], where [tex]\( m = 5 \)[/tex] and [tex]\( n = 1 \)[/tex].

The section formula for the [tex]\( y \)[/tex]-coordinate is given by:
[tex]\[ y = \left(\frac{m}{m+n}\right)(y_2 - y_1) + y_1 \][/tex]

Plugging in the coordinates and the ratio, we have:
[tex]\[ y_1 = -8, \quad y_2 = 6 \][/tex]

First, compute the fraction of the ratio:

[tex]\[ \frac{m}{m+n} = \frac{5}{5+1} = \frac{5}{6} \][/tex]

Next, calculate the difference [tex]\( y_2 - y_1 \)[/tex]:

[tex]\[ y_2 - y_1 = 6 - (-8) = 6 + 8 = 14 \][/tex]

Now, apply these values into the section formula:

[tex]\[ y = \left(\frac{5}{6}\right) \cdot 14 + (-8) \][/tex]

Compute the multiplication:

[tex]\[ \left(\frac{5}{6}\right) \cdot 14 = \frac{70}{6} \approx 11.666666666666668 \][/tex]

Finally, add this result to [tex]\( y_1 \)[/tex]:

[tex]\[ y = 11.666666666666668 - 8 = 3.666666666666668 \][/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]\[ 3.666666666666668 \][/tex]