Question

given points A (-3,-4) and B (2,0), points P (-1,-12/5) partitions line AB in the ratio

Answer 1

$\bf \begin{array}{ccccccccc} &&x_1&&y_1&&x_2&&y_2\\ &A&(~ -3 &,& -4~) &P&(~ -1 &,& -\frac{12}{5}~) \end{array} \\\\\\ d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ AP=\sqrt{[-1-(-3)]^2+[-\frac{12}{5}-(-4)]^2} \\\\\\ AP=\sqrt{(-1+3)^2+(-\frac{12}{5}+4)^2}\implies AP=\sqrt{2^2+\left( \frac{8}{5} \right)^2} \\\\\\ AP=\sqrt{4+\frac{64}{25}}\implies AP=\sqrt{\cfrac{164}{25}}\implies AP=\cfrac{\sqrt{164}}{\sqrt{25}} \\\\\\ \boxed{AP=\cfrac{2\sqrt{41}}{5}}$



$\bf \begin{array}{ccccccccc} &&x_1&&y_1&&x_2&&y_2\\ &P&(~ -1 &,& -\frac{12}{5}~) &B&(~ 2 &,& 0~) \end{array} \\\\\\ PB=\sqrt{[2-(-1)]^2+[0-\left(-\frac{12}{5} \right)]^2} \\\\\\ PB=\sqrt{(2+1)^2+\left( \frac{12}{5} \right)^2}\implies PB=\sqrt{9+\frac{144}{25}}\implies PB=\sqrt{\cfrac{369}{25}} \\\\\\ PB=\cfrac{\sqrt{369}}{\sqrt{25}}\implies \boxed{PB=\cfrac{3\sqrt{41}}{5}}$

so, let's check the ratio of AP:PB

$\bf AP:PB\implies \cfrac{AP}{PB}\implies \cfrac{\frac{2\sqrt{41}}{5}}{\frac{3\sqrt{41}}{5}}\implies \cfrac{2\underline{\sqrt{41}}}{\underline{5}}\cdot \cfrac{\underline{5}}{3\underline{\sqrt{41}}}\implies \cfrac{2}{3}$