Question

the point ( q ) lies on the segment ( overline{pr} ). find the coordinates of ( q ) so that the ratio of ( pq ) to ( qr ) is 7 to 1. ( p(-30, 22) ), ( r(2, -2) ). coordinates of ( q ) : ( (square, square) )

Explanation:

Step1: Recall the section formula

The section formula for a point \( Q(x,y) \) that divides the line segment joining \( P(x_1,y_1) \) 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, \( P(-30,22) \), \( R(2,-2) \), \( m = 7 \), \( n = 1 \).

Step2: Calculate the x-coordinate of Q

Substitute \( x_1=-30 \), \( x_2 = 2 \), \( m = 7 \), \( n = 1 \) into the x - formula:
\[
x=\frac{7\times2+1\times(-30)}{7 + 1}=\frac{14-30}{8}=\frac{-16}{8}=-2
\]

Step3: Calculate the y-coordinate of Q

Substitute \( y_1 = 22 \), \( y_2=-2 \), \( m = 7 \), \( n = 1 \) into the y - formula:
\[
y=\frac{7\times(-2)+1\times22}{7 + 1}=\frac{-14 + 22}{8}=\frac{8}{8}=1
\]

Answer:

\((-2,1)\)