To determine the location of point [tex]\(R\)[/tex] that partitions the directed line segment from [tex]\(Q\)[/tex] to [tex]\(S\)[/tex] in a [tex]\(3:5\)[/tex] ratio, we need to use the formula for the coordinate of a point that divides a segment in a given ratio.
Given:
- [tex]\(Q\)[/tex] is at [tex]\(-14\)[/tex] (let [tex]\(x_1 = -14\)[/tex])
- [tex]\(S\)[/tex] is at [tex]\(2\)[/tex] (let [tex]\(x_2 = 2\)[/tex])
- The ratio is [tex]\(3:5\)[/tex] (let [tex]\(m = 3\)[/tex] and [tex]\(n = 5\)[/tex])
The formula to find the location of point [tex]\(R\)[/tex] is:
[tex]\[
\left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1
\][/tex]
Let's substitute these values into the formula:
[tex]\[
\left(\frac{3}{3+5}\right)(2 - (-14)) + (-14)
\][/tex]
Now check the provided options:
1. [tex]\(\left(\frac{3}{3+5}\right)(2 - (-14)) + (-14)\)[/tex]
2. [tex]\(\left(\frac{3}{3+5}\right)(-14 - 2) + 2\)[/tex]
3. [tex]\(\left(\frac{3}{3+5}\right)(2 - 14) + 14\)[/tex]
4. [tex]\(\left(\frac{3}{3+5}\right)(-14 - 2) - 2\)[/tex]
Among these, the first option is:
[tex]\[
\left(\frac{3}{3+5}\right)(2 - (-14)) + (-14)
\][/tex]
This matches the formula exactly with the given values. Therefore, the correct expression is:
[tex]\[
\left(\frac{3}{3+5}\right)(2 - (-14)) + (-14)
\][/tex]
Solving this we get:
[tex]\[
\left(\frac{3}{8}\right)(16) - 14
\][/tex]
[tex]\[
6 - 14
\][/tex]
[tex]\[
-8
\][/tex]
Thus the correct option is:
[tex]\[
\left(\frac{3}{3+5}\right)(2-(-14))+(-14)
\][/tex]
And the location of point [tex]\(R\)[/tex] is [tex]\(-8\)[/tex].
