To determine the [tex]\( x \)[/tex]-coordinate of the point that divides the directed line segment from 1 to [tex]\( K \)[/tex] into a ratio of [tex]\( 2:5 \)[/tex], I'll outline the steps as follows:
1. Identify the given values:
[tex]\[
m = 2, \quad n = 5, \quad x_1 = 1, \quad x_2 = K
\][/tex]
2. Use the formula for the coordinate dividing a segment in a given ratio:
The [tex]\( x \)[/tex]-coordinate of the point dividing the line segment from [tex]\((x_1)\)[/tex] to [tex]\((x_2)\)[/tex] in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[
x = \left( \frac{m}{m+n} \right)(x_2 - x_1) + x_1
\][/tex]
3. Plug in the values into the ratio formula:
Using [tex]\( m = 2 \)[/tex], [tex]\( n = 5 \)[/tex], and [tex]\( x_1 = 1 \)[/tex]:
[tex]\[
x = \left( \frac{2}{2+5} \right) (K - 1) + 1
\][/tex]
[tex]\[
x = \left( \frac{2}{7} \right) (K - 1) + 1
\][/tex]
4. Determine what [tex]\( K \)[/tex] must be so that [tex]\( x \)[/tex] equals one of the provided choices:
The choices we have for the [tex]\( x \)[/tex]-coordinate are: [tex]\(-4, -2, 2, 4\)[/tex].
5. Check the necessity of having [tex]\( K \)[/tex] to solve numerically:
Since we do not have a numerical value for [tex]\( K \)[/tex], we cannot solve for [tex]\( x \)[/tex] definitively.
- Without a value for [tex]\( K \)[/tex], we are unable to convert the general expression into any of the specific choices given ([tex]\(-4, -2, 2, 4\)[/tex]).
Thus, we conclude:
The [tex]\( x \)[/tex]-coordinate cannot be determined numerically without knowing the value of [tex]\( x_2 = K \)[/tex]. Therefore, none of the provided choices can be confirmed valid.
