Question

the point q lies on the segment $overline{pr}$. find the coordinates of q so that pq is $\frac{4}{9}$ of pr. p (-6, 5) q (?,?) r (21, -13) coordinates of q : ( , )

Explanation:

Step1: Use section - formula for x - coordinate

The section - formula for the x - coordinate of 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 \(x=\frac{mx_2+nx_1}{m + n}\). Here, \(m = 4\), \(n=9 - 4=5\), \(x_1=-6\), and \(x_2 = 21\).
\[x=\frac{4\times21+5\times(-6)}{4 + 5}=\frac{84-30}{9}=\frac{54}{9}=6\]

Step2: Use section - formula for y - coordinate

The section - formula for the y - coordinate is \(y=\frac{my_2+ny_1}{m + n}\). Here, \(y_1 = 5\), \(y_2=-13\), \(m = 4\), and \(n = 5\).
\[y=\frac{4\times(-13)+5\times5}{4 + 5}=\frac{-52 + 25}{9}=\frac{-27}{9}=-3\]

Answer:

\((6,-3)\)