Question

Line segment AB is divided by point P (rounded to the nearest tenth) in the ratio of 1:4. Point A is (7, 5) and point P is (10, 14). What are the coordinates of point B?

Answer 1

`\bf \qquad \textit{internal division of a line segment using ratios}\\\\\\ A(7,5)\qquad B(x,y)\qquad \qquad \stackrel{\textit{ratio from A to B}}{1:4} \\\\\\ \cfrac{A\underline{P}}{\underline{P} B} = \cfrac{1}{4}\implies \cfrac{A}{B} = \cfrac{1}{4}\implies 4A=1B\implies 4(7,5)=1(x,y)\\\\[-0.35em] ~\dotfill\\\\ P=\left(\frac{\textit{sum of`

`\bf P=\left(\cfrac{(4\cdot 7)+(1\cdot x)}{1+4}\quad ,\quad \cfrac{(4\cdot 5)+(1\cdot y)}{1+4}\right)\implies P=\left( \cfrac{28+x}{5}~~,~~\cfrac{20+y}{5} \right) \\\\\\ \stackrel{P}{(10~,~14)}=\left( \cfrac{28+x}{5}~~,~~\cfrac{20+y}{5} \right)\implies \begin{cases} 10=\cfrac{28+x}{5}\\[1em] 50=28+x\\ \boxed{22=x}\\ \cline{1-1} 14=\cfrac{20+y}{5}\\[1em] 70=20+y\\ \boxed{50=y} \end{cases}`