To find the coordinates of point [tex]\( R \)[/tex], which partitions the line segment from [tex]\( Q \)[/tex] to [tex]\( S \)[/tex] in the ratio 4:1, we use the formula
[tex]\[
\left( \frac{m}{m+n} \right) (x_2 - x_1) + x_1
\][/tex]
Given:
- [tex]\( Q \)[/tex] is at [tex]\(-8\)[/tex]
- [tex]\( S \)[/tex] is at [tex]\(12\)[/tex]
- The ratio [tex]\( m:n \)[/tex] is [tex]\(4:1\)[/tex]
Let's identify each variable in the formula:
- [tex]\( x_1 = -8 \)[/tex] (position of [tex]\( Q \)[/tex])
- [tex]\( x_2 = 12 \)[/tex] (position of [tex]\( S \)[/tex])
- [tex]\( m = 4 \)[/tex]
- [tex]\( n = 1 \)[/tex]
Substitute these values into the formula:
[tex]\[
\left( \frac{4}{4+1} \right) (12 - (-8)) + (-8)
\][/tex]
Now, let's break down the calculations step by step:
1. Calculate [tex]\( m + n \)[/tex]:
[tex]\[
4 + 1 = 5
\][/tex]
2. Calculate [tex]\( x_2 - x_1 \)[/tex]:
[tex]\[
12 - (-8) = 12 + 8 = 20
\][/tex]
3. Plug the values back into the formula:
[tex]\[
\left( \frac{4}{5} \right) (20) + (-8)
\][/tex]
4. Simplify the expression inside the parentheses:
[tex]\[
\frac{4 \times 20}{5} = \frac{80}{5} = 16
\][/tex]
5. Add the last component to complete the expression:
[tex]\[
16 + (-8) = 16 - 8 = 8
\][/tex]
So, the location of point [tex]\( R \)[/tex] is [tex]\( 8 \)[/tex].
Among the given options, the correct expression is:
[tex]\[
\left(\frac{4}{4+1}\right)(12-(-8))+(-8)
\][/tex]
Thus, the right formula that correctly finds the location of point [tex]\( R \)[/tex] is [tex]\(\left(\frac{4}{4+1}\right)(12-(-8))+(-8)\)[/tex].
