Question

Find the point P along the directed line segment from point A(-8,-3) to point B(6,12) that divides the segment in the ratio 1 to 7

Answer 1

Bearing in mind that, the ratio here is 1:7 form A to B, that matters, because, the AB segment gets split in 8 pieces, and AP takes 1 slice whilst PB takes 7 slices

$\bf \left. \qquad \right.\textit{internal division of a line segment} \\\\\\ A(-8,-3)\qquad B(6,12)\qquad ratio1=1\qquad ratio2=7\qquad 1:7 \\\\\\ \cfrac{AP}{PB} = \cfrac{1}{7}\implies \cfrac{A}{B} = \cfrac{1}{7}\implies 7\cdot A=1\cdot B \\\\\\ 7(-8,-3)=1(6,12)\\\\ -------------------------------$

$\bf { P=\left(\cfrac{\textit{sum of "x" values}}{ratio1+ratio2}\quad ,\quad \cfrac{\textit{sum of "y" values}}{ratio1+ratio2}\right)}\\\\ -------------------------------\\\\ P=\left(\cfrac{(7\cdot -8)+(1\cdot 6)}{1+7}\quad ,\quad \cfrac{(7\cdot -3)+(1\cdot 12)}{1+7}\right)$