Question

the point k lies on the segment \\(\overline{jl}\\). find the coordinates of k so that the ratio of jk to kl is 5 to 2. l (28,12) k (?,?) j (-7,-2) coordinates of k : (\\(\square\\),\\(\square\\))

Explanation:

Step1: Recall the section formula

The section formula for 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 \( m:n \) is given by:
\[
x=\frac{mx_2 + nx_1}{m + n}, \quad y=\frac{my_2 + ny_1}{m + n}
\]
Here, \( J(-7,-2) \), \( L(28,12) \), \( m = 5 \), \( n = 2 \).

Step2: Calculate the x - coordinate of K

Substitute \( x_1=-7 \), \( x_2 = 28 \), \( m = 5 \), \( n = 2 \) into the formula for \( x \):
\[
x=\frac{5\times28+2\times(-7)}{5 + 2}=\frac{140-14}{7}=\frac{126}{7}=18
\]

Step3: Calculate the y - coordinate of K

Substitute \( y_1=-2 \), \( y_2 = 12 \), \( m = 5 \), \( n = 2 \) into the formula for \( y \):
\[
y=\frac{5\times12+2\times(-2)}{5 + 2}=\frac{60 - 4}{7}=\frac{56}{7}=8
\]

Answer:

\((18, 8)\)