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]\( 3:5 \)[/tex] ratio, we can use the formula:
[tex]\[ R = \left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1 \][/tex]
Given:
- [tex]\( x_1 = -14 \)[/tex] (the coordinate of point [tex]\( Q \)[/tex])
- [tex]\( x_2 = 2 \)[/tex] (the coordinate of point [tex]\( S \)[/tex])
- The ratio [tex]\( 3:5 \)[/tex] translates to [tex]\( m = 3 \)[/tex] and [tex]\( n = 5 \)[/tex]
Substitute these values into the formula:
[tex]\[ R = \left(\frac{3}{3 + 5}\right)(2 - (-14)) + (-14) \][/tex]
First, calculate the fraction:
[tex]\[ \frac{3}{3 + 5} = \frac{3}{8} \][/tex]
Next, compute the difference in the coordinates:
[tex]\[ 2 - (-14) = 2 + 14 = 16 \][/tex]
Now multiply the fraction by this difference:
[tex]\[ \left(\frac{3}{8}\right) \times 16 = \frac{3 \times 16}{8} = 6 \][/tex]
Finally, add this result to [tex]\( x_1 \)[/tex]:
[tex]\[ 6 + (-14) = 6 - 14 = -8 \][/tex]
Therefore, the location of point [tex]\( R \)[/tex] is:
[tex]\[ -8 \][/tex]
The correct expression that uses the formula to find the location of point [tex]\( R \)[/tex] is:
[tex]\[ \left( \frac{3}{3+5} \right)(2 - (-14)) + (-14) \][/tex]
Which matches the first option:
[tex]\[ \left(\frac{3}{3+5}\right)(2-(-14))+(-14) \][/tex]
