To find the [tex]\( x \)[/tex]-coordinate of the point that divides the directed line segment from [tex]\( J \)[/tex] to [tex]\( K \)[/tex] in the ratio [tex]\( 2:5 \)[/tex], we will use the section formula for internal division.
1. Identify the [tex]\( x \)[/tex]-coordinates of points [tex]\( J \)[/tex] and [tex]\( K \)[/tex]:
- Let [tex]\( x_1 \)[/tex] be the [tex]\( x \)[/tex]-coordinate of [tex]\( J \)[/tex], which is [tex]\( -4 \)[/tex].
- Let [tex]\( x_2 \)[/tex] be the [tex]\( x \)[/tex]-coordinate of [tex]\( K \)[/tex], which is [tex]\( 4 \)[/tex].
2. Identify the given ratio:
- The ratio [tex]\( m:n \)[/tex] is given as [tex]\( 2:5 \)[/tex], where [tex]\( m = 2 \)[/tex] and [tex]\( n = 5 \)[/tex].
3. Apply the section formula:
[tex]\[
x = \left( \frac{m}{m+n} \right) (x_2 - x_1) + x_1
\][/tex]
Plugging in the values:
[tex]\[
x = \left( \frac{2}{2+5} \right) (4 - (-4)) + (-4)
\][/tex]
4. Simplify the terms step by step:
- Calculate the sum of the ratio parts:
[tex]\[
m + n = 2 + 5 = 7
\][/tex]
- Calculate the difference between [tex]\( x_2 \)[/tex] and [tex]\( x_1 \)[/tex]:
[tex]\[
x_2 - x_1 = 4 - (-4) = 4 + 4 = 8
\][/tex]
- Substitute these into the formula:
[tex]\[
x = \left( \frac{2}{7} \right) \times 8 + (-4)
\][/tex]
5. Perform the multiplication:
[tex]\[
\left( \frac{2}{7} \right) \times 8 = \frac{16}{7}
\][/tex]
6. Finally, add this value to [tex]\( x_1 \)[/tex]:
[tex]\[
x = \frac{16}{7} - 4
\][/tex]
Convert [tex]\( -4 \)[/tex] to a fraction with the same denominator:
[tex]\[
-4 = -\frac{28}{7}
\][/tex]
Thus, we have:
[tex]\[
x = \frac{16}{7} - \frac{28}{7} = \frac{16 - 28}{7} = \frac{-12}{7} \approx -1.71
\][/tex]
Therefore, the [tex]\( x \)[/tex]-coordinate of the point that divides the directed line segment from [tex]\( J \)[/tex] to [tex]\( K \)[/tex] in the ratio [tex]\( 2:5 \)[/tex] is approximately [tex]\( -1.71 \)[/tex].
