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 2.

p (-3, 4)
q (?, ?)
r (24, -32)
coordinates of q : ( , )

Explanation:

Step1: Recall Section Formula

The section formula for a point \( Q(x,y) \) dividing the line segment joining \( P(x_1,y_1) \) and \( R(x_2,y_2) \) in the ratio \( m:n \) is \( x=\frac{mx_2 + nx_1}{m + n} \), \( y=\frac{my_2 + ny_1}{m + n} \). Here, \( m = 7 \), \( n = 2 \), \( P(-3,4) \), \( R(24,-32) \).

Step2: Calculate x-coordinate of Q

Substitute values into x-formula: \( x=\frac{7\times24 + 2\times(-3)}{7 + 2}=\frac{168 - 6}{9}=\frac{162}{9}=18 \).

Step3: Calculate y-coordinate of Q

Substitute values into y-formula: \( y=\frac{7\times(-32)+2\times4}{7 + 2}=\frac{-224 + 8}{9}=\frac{-216}{9}=-24 \).

Answer:

\((18, -24)\)