To find the point [tex]\( R \)[/tex] that divides the line segment [tex]\( PQ \)[/tex] in the ratio [tex]\( 3:2 \)[/tex], we will use the section formula. The section formula for a point dividing a line segment in a given ratio is given by:
[tex]\[
R_x = \frac{m \cdot Q_x + n \cdot P_x}{m + n}
\][/tex]
[tex]\[
R_y = \frac{m \cdot Q_y + n \cdot P_y}{m + n}
\][/tex]
Where:
- [tex]\( P = (P_x, P_y) = (6, -5) \)[/tex]
- [tex]\( Q = (Q_x, Q_y) = (-2, 4) \)[/tex]
- The ratio [tex]\( m:n = 3:2 \)[/tex]
First, let’s find the x-coordinate of point [tex]\( R \)[/tex]:
[tex]\[
R_x = \frac{3 \cdot (-2) + 2 \cdot 6}{3 + 2}
\][/tex]
Calculate the numerator and the denominator:
[tex]\[
R_x = \frac{(3 \times -2) + (2 \times 6)}{5}
\][/tex]
[tex]\[
R_x = \frac{-6 + 12}{5}
\][/tex]
[tex]\[
R_x = \frac{6}{5}
\][/tex]
[tex]\[
R_x = 1.2
\][/tex]
Next, let’s find the y-coordinate of point [tex]\( R \)[/tex]:
[tex]\[
R_y = \frac{3 \cdot 4 + 2 \cdot (-5)}{3 + 2}
\][/tex]
Calculate the numerator and the denominator:
[tex]\[
R_y = \frac{(3 \times 4) + (2 \times -5)}{5}
\][/tex]
[tex]\[
R_y = \frac{12 - 10}{5}
\][/tex]
[tex]\[
R_y = \frac{2}{5}
\][/tex]
[tex]\[
R_y = 0.4
\][/tex]
So, the coordinates of point [tex]\( R \)[/tex] are [tex]\( (1.2, 0.4) \)[/tex].
Now let's back these coordinates to check which provided option matches:
A. [tex]\( \left(\frac{14}{5},-\frac{7}{5}\right) \)[/tex]
[tex]\[ \approx (2.8, -1.4) \][/tex]
(Doesn't match)
B. [tex]\( \left(\frac{14}{5}, \frac{7}{5}\right) \)[/tex]
[tex]\[ \approx (2.8, 1.4) \][/tex]
(Doesn't match)
C. [tex]\( \left(\frac{6}{5}, \frac{2}{5}\right) \)[/tex]
[tex]\[ \approx (1.2, 0.4) \][/tex]
(Matches)
D. [tex]\( \left(-\frac{6}{5}, \frac{2}{5}\right) \)[/tex]
[tex]\[ \approx (-1.2, 0.4) \][/tex]
(Doesn't match)
Thus, the correct option is:
C. [tex]\(\left(\frac{6}{5}, \frac{2}{5}\right)\)[/tex]
