Question

the point k lies on the segment \\(\overline{jl}\\). find the coordinates of k so that the ratio of jk to kl is 6 to 1. j (-1,1) k (?, ?) l (13, -20)

Explanation:

Step1: Recall the section formula

The section formula for a point \( K(x,y) \) dividing the line segment joining \( J(x_1,y_1) \) and \( L(x_2,y_2) \) 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, \( J(-1,1) \), \( L(13,-20) \), and the ratio \( JK:KL = 6:1 \), so \( m = 6 \), \( n = 1 \), \( x_1=-1 \), \( y_1 = 1 \), \( x_2=13 \), \( y_2=-20 \).

Step2: Calculate the x - coordinate of K

Substitute the values into the formula for \( x \):
\[
x=\frac{6\times13+1\times(-1)}{6 + 1}=\frac{78-1}{7}=\frac{77}{7}=11
\]

Step3: Calculate the y - coordinate of K

Substitute the values into the formula for \( y \):
\[
y=\frac{6\times(-20)+1\times1}{6 + 1}=\frac{-120 + 1}{7}=\frac{-119}{7}=-17
\]

Answer:

The coordinates of \( K \) are \( (11,-17) \)