To solve this problem, we first identify the given elements and apply the correct formula. In this case, we are looking to find the location of point [tex]\( R \)[/tex] which partitions the directed line segment from [tex]\( Q \)[/tex] to [tex]\( S \)[/tex] in a [tex]\( 3:5 \)[/tex] ratio.
1. Identify the endpoints and ratio:
- [tex]\( Q \)[/tex] is at [tex]\(-14\)[/tex]
- [tex]\( S \)[/tex] is at [tex]\(2\)[/tex]
- The ratio [tex]\( m:n \)[/tex] is [tex]\( 3:5 \)[/tex]
2. Set up the formula:
The formula to find the point that partitions the segment in the given ratio is:
[tex]\[
\left(\frac{m}{m+n}\right)\left(x_2 - x_1\right) + x_1
\][/tex]
Where:
- [tex]\( m = 3 \)[/tex]
- [tex]\( n = 5 \)[/tex]
- [tex]\( x_1 = -14 \)[/tex]
- [tex]\( x_2 = 2 \)[/tex]
3. Substitute the values into the formula:
[tex]\[
\left(\frac{3}{3+5}\right)\left(2 - (-14)\right) + (-14)
\][/tex]
Notice that the formula correctly aligns with the expression:
[tex]\[
\left(\frac{3}{3+5}\right)(2 - (-14)) + (-14)
\][/tex]
Therefore, the correct expression is:
[tex]\[
\left(\frac{3}{3+5}\right)(2 - (-14)) + (-14)
\][/tex]
Hence, the correct choice among the given options is:
[tex]\[
\left(\frac{3}{3+5}\right)(2 - (-14)) + (-14)
\][/tex]
