To determine which expression correctly uses the formula [tex]\(\left(\frac{m}{m+n}\right) \left(x_2 - x_1\right) + x_1\)[/tex] for finding the location of point [tex]\(R\)[/tex] that partitions the directed line segment from [tex]\(Q\)[/tex] to [tex]\(S\)[/tex] in a [tex]\(4:1\)[/tex] ratio, let's carefully examine each candidate.
Here, the ratio is given as [tex]\(4:1\)[/tex]. Therefore, [tex]\(m = 4\)[/tex] and [tex]\(n = 1\)[/tex]. The endpoints of the segment are [tex]\(Q = -8\)[/tex] and [tex]\(S = 12\)[/tex], so [tex]\(x_1 = -8\)[/tex] and [tex]\(x_2 = 12\)[/tex].
The formula to find the location of [tex]\(R\)[/tex] is:
[tex]\[
x_R = \left(\frac{m}{m+n}\right) \left(x_2 - x_1\right) + x_1
\][/tex]
Inserting the given values, we have:
[tex]\[
x_R = \left(\frac{4}{4+1}\right) (12 - (-8)) + (-8)
\][/tex]
Let's evaluate the candidates in the given question:
1. [tex]\(\left(\frac{1}{1+4}\right) (12 - (-8)) + (-8)\)[/tex]:
[tex]\[
\left(\frac{1}{5}\right) (12 - (-8)) + (-8) = \left(\frac{1}{5}\right) (20) + (-8) = 4 - 8 = -4
\][/tex]
2. [tex]\(\left(\frac{4}{4+1}\right) (12 - (-8)) + (-8)\)[/tex]:
[tex]\[
\left(\frac{4}{5}\right) (12 - (-8)) + (-8) = \left(\frac{4}{5}\right) (20) + (-8) = 16 - 8 = 8
\][/tex]
3. [tex]\(\left(\frac{4}{4+1}\right) (-8 - 12) + 12\)[/tex]:
[tex]\[
\left(\frac{4}{5}\right) (-8 - 12) + 12 = \left(\frac{4}{5}\right) (-20) + 12 = -16 + 12 = -4
\][/tex]
4. [tex]\(\left(\frac{4}{1+4}\right) (-8 - 12) + 12\)[/tex]:
[tex]\[
\left(\frac{4}{5}\right) (-20) + 12 = -16 + 12 = -4
\][/tex]
To determine which one correctly matches the formula for point [tex]\(R\)[/tex], we observe the results:
- The first option evaluates to [tex]\(-4\)[/tex]
- The second option evaluates to [tex]\(8\)[/tex]
- The third option evaluates to [tex]\(-4\)[/tex]
- The fourth option evaluates to [tex]\(-4\)[/tex]
Based on the calculations, the expression that correctly uses the formula [tex]\(\left(\frac{m}{m+n}\right) \left(x_2 - x_1\right) + x_1\)[/tex] and gives the correct position of point [tex]\(R\)[/tex] is:
[tex]\[
\left(\frac{4}{4+1}\right) (12 - (-8)) + (-8) = 8
\][/tex]
Thus, the correct expression is:
[tex]\[
\left(\frac{4}{4+1}\right)(12-(-8))+(-8)
\][/tex]
