Question

Line segment XY has endpoints X(–10, –1) and Y(5, 15). To find the y-coordinate of the point that divides the directed line segment in a 5:3 ratio, the formula y = (y2 – y1) + y1 was used to find that y = (15 – (–1)) + (–1).

Answer 1

Check the picture below.

so say the point P cuts the segment XY to a ratio of 5:3 from X to Y, thus

$\bf ~~~~~~~~~~~~\textit{internal division of a line segment} \\\\\\ X(-10,-1)\qquad Y(5,15)\qquad \qquad 5:3 \\\\\\ \cfrac{X\underline{P}}{\underline{P}Y} = \cfrac{5}{3}\implies \cfrac{X}{Y} = \cfrac{5}{3}\implies 3X=5Y\implies 3(-10,-1)=5(5,15)\\\\ -------------------------------\\\\ { P=\left(\cfrac{\textit{sum of "x" values}}{\textit{sum of ratios}}\quad ,\quad \cfrac{\textit{sum of "y" values}}{\textit{sum of ratios}}\right)}\\\\ -------------------------------$

$\bf P=\left(\cfrac{(3\cdot -10)+(5\cdot 5)}{5+3}\quad ,\quad \cfrac{(3\cdot -1)+(5\cdot 15)}{5+3}\right) \\\\\\ P=\left( \cfrac{-30+25}{8}~~,~~\cfrac{-3+75}{8} \right)\implies P=\left( \cfrac{-5}{8}~~,~~\cfrac{72}{8} \right) \\\\\\ P=\left( -\frac{5}{8}~~,~~ 9\right)$