What is 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]?

[tex]\[ x = \left(\frac{m}{m+n}\right)\left(x_2 - x_1\right) + x_1 \][/tex]

A. \(-1\)
B. 3
C. 7
D. 11



Answer :

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]