To find the \( x \)-coordinate of the point dividing the line segment from \( K \) to \( J \) in the ratio \( 1:3 \), we can use the section formula.
Given:
- Coordinates of point \( K \) are \( (-1, \cdot) \)
- Coordinates of point \( J \) are \( (11, \cdot) \)
- Ratio \( m:n \) is \( 1:3 \)
The section formula for the \( x \)-coordinate is:
[tex]\[
x = \left(\frac{m}{m+n}\right) \left(x_2 - x_1\right) + x_1
\][/tex]
Let's substitute the given values into the formula:
- \( x_1 = -1 \)
- \( x_2 = 11 \)
- \( m = 1 \)
- \( n = 3 \)
First, compute the fraction for the ratio:
[tex]\[
\frac{m}{m+n} = \frac{1}{1+3} = \frac{1}{4}
\][/tex]
Next, compute \( x_2 - x_1 \):
[tex]\[
x_2 - x_1 = 11 - (-1) = 11 + 1 = 12
\][/tex]
Now, multiply the fraction by the difference:
[tex]\[
\frac{1}{4} \times 12 = 3
\][/tex]
Finally, add \( x_1 \):
[tex]\[
x = 3 + (-1) = 2
\][/tex]
Thus, the \( x \)-coordinate of the point that divides the directed line segment from \( K \) to \( J \) in the ratio \( 1:3 \) is:
[tex]\[
\boxed{2}
\][/tex]
