Question

1. given (overline{jl}) with (j(8, -8)) and (l(-16, -2)), find the coordinates of (k) if (k) divides (jl) two - thirds of the way from (j) to (l).

Explanation:

Step1: Recall the section formula

The coordinates of a point \( K(x,y) \) that divides the line segment joining \( J(x_1,y_1) \) and \( L(x_2,y_2) \) 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, \( K \) divides \( JL \) two - thirds of the way from \( J \) to \( L \), so the ratio \( m:n = 2:1 \) (since the distance from \( J \) to \( K \) is \( \frac{2}{3} \) of \( JL \), and from \( K \) to \( L \) is \( \frac{1}{3} \) of \( JL \)). Given \( J(8,-8)=(x_1,y_1) \) and \( L(-16,-2)=(x_2,y_2) \), \( m = 2 \), \( n=1 \).

Step2: Calculate the x - coordinate of K

Using the formula for the x - coordinate: \( x=\frac{mx_2+nx_1}{m + n} \)
Substitute \( m = 2 \), \( n = 1 \), \( x_1=8 \), \( x_2=-16 \)
\( x=\frac{2\times(-16)+1\times8}{2 + 1}=\frac{-32 + 8}{3}=\frac{-24}{3}=-8 \)

Step3: Calculate the y - coordinate of K

Using the formula for the y - coordinate: \( y=\frac{my_2+ny_1}{m + n} \)
Substitute \( m = 2 \), \( n = 1 \), \( y_1=-8 \), \( y_2=-2 \)
\( y=\frac{2\times(-2)+1\times(-8)}{2 + 1}=\frac{-4-8}{3}=\frac{-12}{3}=-4 \)

Answer:

The coordinates of \( K \) are \( (-8,-4) \)