Question

the coordinates of the endpoints of \\( \overline{qr} \\) are \\( q(-11, -12) \\) and \\( r(3, 2) \\). point \\( s \\) is on \\( \overline{qr} \\) and divides it such that \\( qs:rs \\) is \\( 5:2 \\). what are the coordinates of \\( s \\)? write your answers as integers or decimals.

Explanation:

Step1: Recall the section formula

The section formula for a point \( S(x,y) \) that divides the line segment joining \( Q(x_1,y_1) \) and \( R(x_2,y_2) \) 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, \( Q(-11,-12) \), \( R(3,2) \), \( m = 5 \), \( n = 2 \).

Step2: Calculate the x - coordinate of S

Substitute \( x_1=-11 \), \( x_2 = 3 \), \( m = 5 \), \( n = 2 \) into the formula for \( x \):
\[
x=\frac{5\times3+2\times(-11)}{5 + 2}=\frac{15-22}{7}=\frac{-7}{7}=-1
\]

Step3: Calculate the y - coordinate of S

Substitute \( y_1=-12 \), \( y_2 = 2 \), \( m = 5 \), \( n = 2 \) into the formula for \( y \):
\[
y=\frac{5\times2+2\times(-12)}{5 + 2}=\frac{10-24}{7}=\frac{-14}{7}=-2
\]

Answer:

\((-1, -2)\)