To determine which expression correctly uses the formula [tex]\(\left(\frac{m}{m+n}\right)\left(x_2 - x_1\right) + x_1\)[/tex] to find the location of point [tex]\(R\)[/tex], let's break down the problem step-by-step.
1. Identify the given values:
- Endpoint [tex]\( Q \)[/tex] is at -8.
- Endpoint [tex]\( S \)[/tex] is at 12.
- The ratio [tex]\( m : n \)[/tex] is [tex]\( 4 : 1 \)[/tex], where [tex]\( m = 4 \)[/tex] and [tex]\( n = 1 \)[/tex].
2. Set up the formula:
The formula to find the point [tex]\( R \)[/tex] that partitions the segment [tex]\([Q, S]\)[/tex] in the ratio [tex]\( m : n \)[/tex] is:
[tex]\[
R = \left( \frac{m}{m+n} \right) (x_2 - x_1) + x_1
\][/tex]
3. Substitute the values into the formula:
- In our case, [tex]\( x_1 = Q = -8 \)[/tex],
- [tex]\( x_2 = S = 12 \)[/tex],
- [tex]\( m = 4 \)[/tex],
- [tex]\( n = 1 \)[/tex].
Plugging these values into the formula, we get:
[tex]\[
R = \left( \frac{4}{4+1} \right) (12 - (-8)) + (-8)
\][/tex]
4. Check the given expressions:
- [tex]\(\left(\frac{1}{1+4}\right)(12-(-8))+(-8)\)[/tex]
- [tex]\(\left(\frac{4}{4+1}\right)(12-(-8))+(-8)\)[/tex]
- [tex]\(\left(\frac{4}{4+1}\right)(-8-12)+12\)[/tex]
- [tex]\(\left(\frac{4}{1+4}\right)(-8-12)+12\)[/tex]
Comparing these with our substituted formula:
[tex]\[
\left( \frac{4}{4+1} \right) (12 - (-8)) + (-8)
\][/tex]
The correct expression matches the second option:
[tex]\[
\left(\frac{4}{4+1}\right)(12-(-8))+(-8)
\][/tex]
Therefore, the correct expression is:
[tex]\[
\left(\frac{4}{4+1}\right)(12-(-8))+(-8)
\][/tex]
