To find 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, we need to use the formula:
[tex]\[
R = \left( \frac{m}{m+n} \right) (x_2 - x_1) + x_1
\][/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.
Given:
- [tex]\( Q = -8 \)[/tex]
- [tex]\( S = 12 \)[/tex]
- [tex]\( m = 4 \)[/tex]
- [tex]\( n = 1 \)[/tex]
Substituting these values into the formula, we get:
[tex]\[
R = \left( \frac{4}{4+1} \right) (12 - (-8)) + (-8)
\][/tex]
Let's identify the expression that matches this substitution:
It is clearly:
[tex]\[
\left( \frac{4}{4 + 1} \right) (12 - (-8)) + (-8)
\][/tex]
To summarize, the correct expression that uses the formula to find the location of point [tex]\( R \)[/tex] is:
[tex]\[
\boxed{\left( \frac{4}{4+1} \right) (12 - (-8)) + (-8)}
\][/tex]
Which is the second option:
[tex]\[\left(\frac{4}{4+1}\right)(12-(-8))+(-8)\][/tex]
