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 follow these steps:
1. Identify the coordinates:
- Point [tex]\( Q \)[/tex] is at [tex]\( x_1 = -8 \)[/tex].
- Point [tex]\( S \)[/tex] is at [tex]\( x_2 = 12 \)[/tex].
- The ratio is [tex]\( m:n = 4:1 \)[/tex], where [tex]\( m = 4 \)[/tex] and [tex]\( n = 1 \)[/tex].
2. Substitute the values into the formula:
The formula for finding point [tex]\( R \)[/tex] on the line segment from [tex]\( Q \)[/tex] to [tex]\( S \)[/tex] is:
[tex]\[
R = \left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1
\][/tex]
3. Plug in the given numbers:
[tex]\[
R = \left(\frac{4}{4+1}\right)(12 - (-8)) + (-8)
\][/tex]
4. Simplify inside the parentheses first:
[tex]\[
4 + 1 = 5
\][/tex]
[tex]\[
12 - (-8) = 12 + 8 = 20
\][/tex]
5. Perform the division and the multiplication:
[tex]\[
\frac{4}{5} \times 20 = \frac{4 \times 20}{5} = \frac{80}{5} = 16
\][/tex]
6. Add this result to [tex]\( x_1 \)[/tex]:
[tex]\[
16 + (-8) = 16 - 8 = 8
\][/tex]
Therefore, the location of point [tex]\( R \)[/tex] is [tex]\( 8 \)[/tex].
The correct expression that gives this result is:
[tex]\[
\left(\frac{4}{4+1}\right)(12-(-8))+(-8)
\][/tex]
So, the answer is the second option:
[tex]\[
\left(\frac{4}{4+1}\right)(12-(-8))+(-8)
\][/tex]
