To determine the correct location of point [tex]\( R \)[/tex] that partitions the directed line segment from [tex]\( Q \)[/tex] to [tex]\( S \)[/tex] in the ratio 3:5, we will use the formula for finding the coordinates of a point dividing a segment in a given ratio:
[tex]\[ x_R = \left( \frac{m}{m + n} \right)(x_2 - x_1) + x_1 \][/tex]
Here, the points and ratios are:
- [tex]\( Q \)[/tex] has coordinates [tex]\( x_1 = -14 \)[/tex]
- [tex]\( S \)[/tex] has coordinates [tex]\( x_2 = 2 \)[/tex]
- The ratio in which [tex]\( R \)[/tex] divides [tex]\( QS \)[/tex] is [tex]\( 3:5 \)[/tex], where [tex]\( m = 3 \)[/tex] and [tex]\( n = 5 \)[/tex]
By substituting these values into the formula:
[tex]\[ x_R = \left( \frac{3}{3+5} \right)(2 - (-14)) + (-14) \][/tex]
Let's break this down step by step:
1. Calculate the sum of the ratio components:
[tex]\[ 3 + 5 = 8 \][/tex]
2. Substitute the ratio into the fraction:
[tex]\[ \frac{3}{8} \][/tex]
3. Compute the difference between [tex]\( x_2 \)[/tex] (the coordinate of [tex]\( S \)[/tex]) and [tex]\( x_1 \)[/tex] (the coordinate of [tex]\( Q \)[/tex]):
[tex]\[ 2 - (-14) = 2 + 14 = 16 \][/tex]
4. Multiply the fraction by the difference calculated in step 3:
[tex]\[ \frac{3}{8} \times 16 = 6 \][/tex]
5. Finally, add this result to [tex]\( x_1 \)[/tex] (the coordinate of [tex]\( Q \)[/tex]):
[tex]\[ 6 + (-14) = -8 \][/tex]
So, the location of point [tex]\( R \)[/tex] is [tex]\(-8\)[/tex].
Thus, the correct expression that correctly uses the formula is:
[tex]\[ \left( \frac{3}{3+5} \right)(2-(-14)) + (-14) \][/tex]
Which matches the option:
[tex]\[ \left( \frac{3}{3+5} \right)(2 - (-14)) + (-14) \][/tex]
