Question

Line segment AB has endpoints A(1, 4) and B(6, 2). Find the coordinates of the point that divides the line segment directed from A to B in the ratio of 2:3.

Answer 1

let's say that point is point C, thus

$\bf ~~~~~~~~~~~~\textit{internal division of a line segment}\\\\\\A(1,4)\qquad B(6,2)\qquad\qquad \stackrel{\textit{ratio from A to B}}{2:3}\\\\\\ \cfrac{A\underline{C}}{\underline{C} B} = \cfrac{2}{3}\implies \cfrac{A}{B} = \cfrac{2}{3}\implies 3A=2B\implies 3(1,4)=2(6,2)\\\\[-0.35em]~\dotfill\\\\C=\left(\frac{\textit{sum of "x" values}}{\textit{sum of ratios}}\quad ,\quad \frac{\textit{sum of "y" values}}{\textit{sum of ratios}}\right)\\\\[-0.35em]~\dotfill$

$\bf C=\left(\cfrac{(3\cdot 1)+(2\cdot 6)}{2+3}\quad ,\quad \cfrac{(3\cdot 4)+(2\cdot 2)}{2+3}\right)\implies C=\left(\cfrac{3+12}{5}~,~\cfrac{12+4}{5} \right)\\\\\\C=\left( \cfrac{15}{5}~,~\cfrac{16}{5} \right)\implies C=\left( 3~,~3\frac{1}{5} \right)$

Answer 2

Answer:

D. (  24/5,  19/5)

Step-by-step explanation:

(mx2 + nx1)            (my2 + ny1)

  (m + n)            ,        (m + n)

Where the point divides the segment internally in the ratio m:n

((2)(3) + (3)(6))               ((2)(8) + (3)(1))  

     (2 + 3)              ,             (2 + 3)                = 24/5, 19/5