Question

Points P, Q, R and S divide a line segment joining A (2, 6) and B (7, -4) in five equal parts. Find the coordinates of P and R. :-;​

Answer 1

Check the picture below.

so we can look at this, this way, let's find P which is 1/5 of the way from A to B, and also find R which is 3/5 of the way from A to B.

`\textit{internal division of a segment using a fraction}\\\\ A(\stackrel{x_1}{2}~,~\stackrel{y_1}{6})\qquad B(\stackrel{x_2}{7}~,~\stackrel{y_2}{-4})~\hspace{8em} (1)/(5)\textit{ of the way from A to B} \\\\[-0.35em] ~\dotfill\\\\ (\stackrel{x_2}{7}-\stackrel{x_1}{2}~~,~~ \stackrel{y_2}{-4}-\stackrel{y_1}{6})\qquad \implies \qquad \stackrel{\stackrel{\textit{component form of}}{\textit{segment AB}}}{\left( 5 ~~,~~ -10 \right)} \\\\[-0.35em] ~\dotfill`

`\left( \stackrel{x_1}{2}~~+~~(1)/(5)(5)~~,~~\stackrel{y_1}{6}~~+~~(1)/(5)(-10) \right)\implies \stackrel{\textit{\LARGE P} }{(3~~,~~4)} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \textit{internal division of a segment using a fraction}\\\\ A(\stackrel{x_1}{2}~,~\stackrel{y_1}{6})\qquad B(\stackrel{x_2}{7}~,~\stackrel{y_2}{-4})~\hspace{8em} (3)/(5)\textit{ of the way from A to B} \\\\[-0.35em] ~\dotfill`

`(\stackrel{x_2}{7}-\stackrel{x_1}{2}~~,~~ \stackrel{y_2}{-4}-\stackrel{y_1}{6})\qquad \implies \qquad \stackrel{\stackrel{\textit{component form of}}{\textit{segment AB}}}{\left( 5 ~~,~~ -10 \right)} \\\\[-0.35em] ~\dotfill\\\\ \left( \stackrel{x_1}{2}~~+~~(3)/(5)(5)~~,~~\stackrel{y_1}{6}~~+~~(3)/(5)(-10) \right)\implies \stackrel{ \textit{\LARGE R} }{(5~~,~~0)}`