Question

Find the coordinates of the point three tenths

of the way from A(-4, -8) to B(11, 7)

Answer 1

let's first off take a peek at those values.

let's say the point with those coordinates is point C, so C is 3/10 of the way from A to B.

meaning, we take the segment AB and cut it in 10 equal pieces, AC takes 3 pieces, and CB takes 7 pieces, namely AC and CB are at a 3:7 ratio.

`\bf ~~~~~~~~~~~~\textit{internal division of a line segment} \\\\\\ A(-4,-8)\qquad B(11,7)\qquad \qquad \stackrel{\textit{ratio from A to B}}{3:7} \\\\\\ \cfrac{A\underline{C}}{\underline{C} B} = \cfrac{3}{7}\implies \cfrac{A}{B} = \cfrac{3}{7}\implies 7A=3B\implies 7(-4,-8)=3(11,7)\\\\[-0.35em] ~\dotfill\\\\ C=\left(\frac{\textit{sum of`

`\bf C=\left(\cfrac{(7\cdot -4)+(3\cdot 11)}{3+7}\quad ,\quad \cfrac{(7\cdot -8)+(3\cdot 7)}{3+7}\right) \\\\\\ C=\left( \cfrac{-28+33}{10}~~,~~\cfrac{-56+21}{10} \right)\implies C=\left( \cfrac{5}{10}~~,~~\cfrac{-35}{10} \right) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill C=\left( (1)/(2)~,~-(7)/(2) \right)~\hfill`