To determine the [tex]\( x \)[/tex]-coordinate of the point that divides the directed line segment from [tex]\( K \)[/tex] to [tex]\( J \)[/tex] in the ratio [tex]\( 1:3 \)[/tex], you can use the section formula. Here is the step-by-step solution:
1. Identify the coordinates and the given ratio:
- The coordinates of [tex]\( K \)[/tex] and [tex]\( J \)[/tex] are given as [tex]\( x_1 = -1 \)[/tex] and [tex]\( x_2 = 3 \)[/tex], respectively.
- The ratio [tex]\( m:n \)[/tex] is [tex]\( 1:3 \)[/tex].
2. Substitute [tex]\( m \)[/tex] and [tex]\( n \)[/tex]:
- Here, [tex]\( m = 1 \)[/tex] and [tex]\( n = 3 \)[/tex].
3. Plug the values into the section formula:
The formula to find the [tex]\( x \)[/tex]-coordinate that divides a line segment in a given ratio is:
[tex]\[
x = \left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1
\][/tex]
Substituting the values:
[tex]\[
x = \left(\frac{1}{1+3}\right)(3 - (-1)) + (-1)
\][/tex]
4. Simplify the terms inside the formula:
- Calculate the denominator [tex]\( m + n \)[/tex]:
[tex]\[
1 + 3 = 4
\][/tex]
- Calculate the difference [tex]\( x_2 - x_1 \)[/tex]:
[tex]\[
3 - (-1) = 3 + 1 = 4
\][/tex]
- Now, substitute these values back into the formula to get:
[tex]\[
x = \left(\frac{1}{4}\right) \cdot 4 - 1
\][/tex]
5. Perform the multiplication:
[tex]\[
x = 1 - 1 = 0
\][/tex]
6. Conclude the solution:
Therefore, the [tex]\( x \)[/tex]-coordinate of the point that divides the directed line segment from [tex]\( K \)[/tex] to [tex]\( J \)[/tex] into a ratio of [tex]\( 1:3 \)[/tex] is:
[tex]\[
\boxed{0.0}
\][/tex]
The [tex]\( x \)[/tex]-coordinate that divides the line segment in the given ratio is indeed [tex]\( 0.0 \)[/tex].
