To find the [tex]\( x \)[/tex]-coordinate of the point that divides the directed line segment from point [tex]\( K \)[/tex] to point [tex]\( J \)[/tex] in the ratio of [tex]\( 1: 3 \)[/tex], we can use the section formula. This formula is given by:
[tex]\[ x = \left(\frac{m}{m+n}\right) \left(x_2 - x_1\right) + x_1 \][/tex]
Here, the variables represent the following:
- [tex]\( m \)[/tex] and [tex]\( n \)[/tex] are the parts of the ratio (in this case, [tex]\( m = 1 \)[/tex] and [tex]\( n = 3 \)[/tex]),
- [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex] are the [tex]\( x \)[/tex]-coordinates of points [tex]\( K \)[/tex] and [tex]\( J \)[/tex], respectively.
Given:
[tex]\[ m = 1 \][/tex]
[tex]\[ n = 3 \][/tex]
[tex]\[ x_1 = -1 \][/tex]
[tex]\[ x_2 = 7 \][/tex]
Let's plug these values into the section formula:
[tex]\[ x = \left(\frac{1}{1+3}\right) \left(7 - (-1)\right) + (-1) \][/tex]
First, add the values in the ratio:
[tex]\[ m + n = 1 + 3 = 4 \][/tex]
Next, calculate the difference in the [tex]\( x \)[/tex]-coordinates of points [tex]\( J \)[/tex] and [tex]\( K \)[/tex]:
[tex]\[ x_2 - x_1 = 7 - (-1) = 7 + 1 = 8 \][/tex]
Now, multiply the ratio by this difference:
[tex]\[ \left(\frac{1}{4}\right) \times 8 = \frac{8}{4} = 2 \][/tex]
Finally, add this result to [tex]\( x_1 \)[/tex]:
[tex]\[ x = 2 + (-1) = 2 - 1 = 1 \][/tex]
Therefore, the [tex]\( x \)[/tex]-coordinate of the point that divides the segment from [tex]\( K \)[/tex] to [tex]\( J \)[/tex] in the ratio of [tex]\( 1: 3 \)[/tex] is:
[tex]\[ x = 1.0 \][/tex]
