Sure, let's solve this step by step.
### Given:
- Endpoint [tex]\(Q\)[/tex] at [tex]\(-8\)[/tex]
- Endpoint [tex]\(S\)[/tex] at [tex]\(12\)[/tex]
- Point [tex]\(R\)[/tex] partitions the segment [tex]\(QS\)[/tex] in a [tex]\(4:1\)[/tex] ratio
We're asked to find the expression that correctly uses the formula to locate point [tex]\(R\)[/tex]. The correct formula to use is:
[tex]\[
\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1
\][/tex]
where:
- [tex]\(x_1\)[/tex] is the coordinate of point [tex]\(Q\)[/tex]
- [tex]\(x_2\)[/tex] is the coordinate of point [tex]\(S\)[/tex]
- [tex]\(m\)[/tex] and [tex]\(n\)[/tex] are the ratios in which [tex]\(R\)[/tex] partitions [tex]\(QS\)[/tex]
### Step-by-step Solution:
1. Identify the coordinates:
- [tex]\(x_1 = -8\)[/tex] (coordinate of [tex]\(Q\)[/tex])
- [tex]\(x_2 = 12\)[/tex] (coordinate of [tex]\(S\)[/tex])
2. Identify the ratio values:
- [tex]\(m = 4\)[/tex]
- [tex]\(n = 1\)[/tex]
3. Substitute the values into the formula:
[tex]\[
\left(\frac{4}{4+1}\right)\left(12 - (-8)\right) + (-8)
\][/tex]
4. Simplify the expression inside the formula:
[tex]\[
\left(\frac{4}{5}\right)(12 + 8) - 8
\][/tex]
Thus, the correct expression is:
[tex]\[
\left(\frac{4}{4+1}\right)(12-(-8)) + (-8)
\][/tex]
This matches the second option provided:
[tex]\[
\left(\frac{4}{4+1}\right)(12-(-8))+(-8)
\][/tex]
Hence, the correct expression is:
\[
\left(\frac{4}{4+1}\right)(12-(-8))+(-8)
\
