Question

1. given \\(\overline{wy}\\) with \\(w(3, 7)\\) and \\(y(13, -8)\\), if \\(x\\) partitions \\(wy\\) such that the ratio of \\(wx\\) to \\(xy\\) is \\(3:2\\), find the coordinates of \\(x\\).

Explanation:

Step1: Recall the section formula

The section formula for a point \( X(x,y) \) that divides the line segment joining \( W(x_1,y_1) \) and \( Y(x_2,y_2) \) in the ratio \( m:n \) is given by:
\[
x=\frac{mx_2 + nx_1}{m + n}, \quad y=\frac{my_2 + ny_1}{m + n}
\]
Here, \( W(3,7) \), \( Y(13,-8) \), \( m = 3 \), \( n = 2 \).

Step2: Calculate the x-coordinate of X

Substitute \( x_1 = 3 \), \( x_2 = 13 \), \( m = 3 \), \( n = 2 \) into the x - formula:
\[
x=\frac{3\times13+2\times3}{3 + 2}=\frac{39+6}{5}=\frac{45}{5}=9
\]

Step3: Calculate the y-coordinate of X

Substitute \( y_1 = 7 \), \( y_2=-8 \), \( m = 3 \), \( n = 2 \) into the y - formula:
\[
y=\frac{3\times(-8)+2\times7}{3 + 2}=\frac{-24 + 14}{5}=\frac{-10}{5}=-2
\]

Answer:

The coordinates of \( X \) are \( (9,-2) \)