Question

the point q lies on the segment \\( \overline{pr} \\).
find the coordinates of q so that the ratio of pq to qr is 3 to 2.
image of a segment with p(-13, 18), r(2, -7), and q(x, y) marked
coordinates of q : (\\( \square \\), \\( \square \\))

Explanation:

Step1: Recall the 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 \( x=\frac{mx_2 + nx_1}{m + n} \) and \( y=\frac{my_2 + ny_1}{m + n} \). Here, \( m = 3 \), \( n = 2 \), \( P(-13,18) \), \( R(2,-7) \).

Step2: Calculate the x - coordinate of Q

Substitute \( m = 3 \), \( n = 2 \), \( x_1=-13 \), \( x_2 = 2 \) into the x - coordinate formula:
\( x=\frac{3\times2+2\times(-13)}{3 + 2}=\frac{6-26}{5}=\frac{-20}{5}=-4 \)

Step3: Calculate the y - coordinate of Q

Substitute \( m = 3 \), \( n = 2 \), \( y_1 = 18 \), \( y_2=-7 \) into the y - coordinate formula:
\( y=\frac{3\times(-7)+2\times18}{3 + 2}=\frac{-21 + 36}{5}=\frac{15}{5}=3 \)

Answer:

\((-4,3)\)