Question

For the line segment whose end points are R(1,2) and S(6,7) find the y value for the point located 3/4 the distance from R to S.

Answer 1

So, let's see, the point say P, is 3/4 of the way from R to S, namely, if we split the segment RS into 4 pieces, from R to P, or RP will take 3 of those quarters, and from P to S, or PS, will take one of those quarters, check the picture below.

so the RP section is at a ratio of 3, whilst the PS section is at a ratio of 1.

$\bf ~~~~~~~~~~~~\textit{internal division of a line segment} \\\\\\ R(1,2)\qquad S(6,7)\qquad \qquad 3:1 \\\\\\ \cfrac{R\underline{P}}{\underline{P} S} = \cfrac{3}{1}\implies \cfrac{R}{S} = \cfrac{3}{1}\implies 1R=3S\implies 1(1,2)=3(6,7)\\\\ -------------------------------$

$\bf { P=\left(\cfrac{\textit{sum of "x" values}}{\textit{sum of ratios}}\quad ,\quad \cfrac{\textit{sum of "y" values}}{\textit{sum of ratios}}\right)}\\\\ -------------------------------\\\\ P=\left(\cfrac{(1\cdot 1)+(3\cdot 6)}{3+1}\quad ,\quad \cfrac{(1\cdot 2)+(3\cdot 7)}{3+1}\right) \\\\\\ P=\left( \qquad \qquad ,\cfrac{2+21}{4} \right)\implies P=\left( \qquad \qquad ,\cfrac{23}{4} \right)$

and that's the y-coordinate for the point P.