On a number line, the directed line segment from [tex]\( Q \)[/tex] to [tex]\( S \)[/tex] has endpoints [tex]\( Q \)[/tex] at -8 and [tex]\( S \)[/tex] at 12. Point [tex]\( R \)[/tex] partitions the directed line segment from [tex]\( Q \)[/tex] to [tex]\( S \)[/tex] in a [tex]\( 4:1 \)[/tex] ratio.

Which expression correctly uses the formula [tex]\(\left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1\)[/tex] to find the location of point [tex]\( R \)[/tex]?

A. [tex]\(\left(\frac{1}{1+4}\right)(12 - (-8)) + (-8)\)[/tex]

B. [tex]\(\left(\frac{4}{4+1}\right)(12 - (-8)) + (-8)\)[/tex]

C. [tex]\(\left(\frac{4}{4+1}\right)(-8 - 12) + 12\)[/tex]

D. [tex]\(\left(\frac{4}{1+4}\right)(-8 - 12) + 12\)[/tex]



Answer :

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]