Question

1. given \\(\overline{xz}\\) with \\(x(-4, 3)\\) and \\(z(6, -2)\\), find the coordinates of \\(y\\) if \\(y\\) divides \\(xz\\) one-fifth of the way from \\(x\\) to \\(z\\).

Explanation:

Step1: Recall the section formula

The coordinates of a point \( Y(x,y) \) that divides the line segment joining \( X(x_1,y_1) \) and \( Z(x_2,y_2) \) in the ratio \( m:n \) are given by \( x=\frac{mx_2 + nx_1}{m + n} \) and \( y=\frac{my_2 + ny_1}{m + n} \). Here, \( Y \) divides \( XZ \) one - fifth of the way from \( X \) to \( Z \), so the ratio \( m:n = 1:4 \) (since the total number of parts is 5, and \( Y \) is 1 part from \( X \) and \( 5 - 1=4 \) parts from \( Z \)). Given \( X(-4,3) \) so \( x_1=-4,y_1 = 3 \) and \( Z(6,-2) \) so \( x_2 = 6,y_2=-2 \), \( m = 1 \), \( n = 4 \).

Step2: Calculate the x - coordinate of Y

Using the formula for the x - coordinate: \( x=\frac{mx_2+nx_1}{m + n} \)
Substitute \( m = 1 \), \( n = 4 \), \( x_1=-4 \), \( x_2 = 6 \)
\( x=\frac{1\times6+4\times(-4)}{1 + 4}=\frac{6-16}{5}=\frac{-10}{5}=-2 \)

Step3: Calculate the y - coordinate of Y

Using the formula for the y - coordinate: \( y=\frac{my_2+ny_1}{m + n} \)
Substitute \( m = 1 \), \( n = 4 \), \( y_1 = 3 \), \( y_2=-2 \)
\( y=\frac{1\times(-2)+4\times3}{1 + 4}=\frac{-2 + 12}{5}=\frac{10}{5}=2 \)

Answer:

The coordinates of \( Y \) are \( (-2,2) \)