Question

What is 3/10 of the way from (-3,-6) to (9,4)

Answer 1

we'll first off put the two points in component form, then we'll multiply that by the fraction 3/10 and those coordinates we'll simply add to the first point, anyhow, let's proceed,

$\bf \textit{internal division of a segment using a fraction}\\\\ A(\stackrel{x_1}{-3}~,~\stackrel{y_1}{-6})\qquad B(\stackrel{x_2}{9}~,~\stackrel{y_2}{4})~\hspace{8em} \frac{3}{10}\textit{ of the way from }A\to B \\\\[-0.35em] ~\dotfill\\\\ \stackrel{~\hfill \textit{component form of segment AB}}{ (\stackrel{x_2}{9}-\stackrel{x_1}{(-3)}, \stackrel{y_2}{4}-\stackrel{y_1}{(-6)})\implies (9+3~,~4+6)\qquad \implies \qquad (12,10)} \\\\[-0.35em] ~\dotfill$

$\bf \stackrel{~\hfill \textit{values to be added to Point A coordinates}} {\cfrac{3}{10}(12,10)\implies \cfrac{3}{10}(12)~,~\cfrac{3}{10}(10)\qquad \implies \qquad \left(\frac{18}{5}~,~3\right)} \\\\\\ \stackrel{\textit{addition of Point A coordinates and values}} {A(-3,-6)+\left( \frac{18}{5},3\right)\implies \left( -3+\frac{18}{5}~~,~~-6+3 \right)\qquad \implies \left( \frac{3}{5}~,~-3\right)}$