Question

line segment pq is a directed line segment beginning at p(6, - 5) and ending at q(- 2,4). find point r on the line segment pq that partitions it into the segments pr and rq in the ratio 3:2. a. $(\frac{14}{5},\frac{7}{5})$ b. $(-\frac{8}{5},\frac{2}{5})$ c. $(\frac{8}{5},\frac{2}{5})$ d. $(\frac{14}{5},-\frac{7}{5})$

Explanation:

Step1: Use the section formula

If a point \(R(x,y)\) divides the line segment joining \(P(x_1,y_1)\) and \(Q(x_2,y_2)\) in the ratio \(m:n\), then \(x=\frac{mx_2 + nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(x_1 = 6,y_1=-5,x_2=-2,y_2 = 4,m = 3,n = 2\).

Step2: Calculate the \(x\) - coordinate of \(R\)

Substitute the values into the \(x\) - coordinate formula:
\(x=\frac{3\times(-2)+2\times6}{3 + 2}=\frac{-6 + 12}{5}=\frac{6}{5}\)

Step3: Calculate the \(y\) - coordinate of \(R\)

Substitute the values into the \(y\) - coordinate formula:
\(y=\frac{3\times4+2\times(-5)}{3 + 2}=\frac{12-10}{5}=\frac{2}{5}\)

Answer:

C. \((\frac{6}{5},\frac{2}{5})\)