so say hmmm the point C is 3/5 of the way from A to B, that means the segment AC is at a ratio of 3, whilst the segment CB is at a ratio of 5, namely 3:5, and C cuts AB to a 3:5 ratio, namely is 3/5 of the way from A to B, check the picture below. Thus,
[tex]\bf \left. \qquad \right.\textit{internal division of a line segment}
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A(-9,3)\qquad B(21,2)\qquad
\qquad 3:5
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\cfrac{AC}{CB} = \cfrac{3}{5}\implies \cfrac{A}{B} = \cfrac{3}{5}\implies 5A=3B\implies 5(-9,3)=3(21,2)\\\\
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{ C=\left(\cfrac{\textit{sum of "x" values}}{\textit{sum of ratios}}\quad ,\quad \cfrac{\textit{sum of "y" values}}{\textit{sum of ratios}}\right)}\\\\
-------------------------------[/tex]
[tex]\bf C=\left(\cfrac{(5\cdot -9)+(3\cdot 21)}{3+5}\quad ,\quad \cfrac{(5\cdot 3)+(3\cdot 2)}{3+5}\right)
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C=\left(\cfrac{-45+63}{8}~~,~~\cfrac{15+6}{8} \right)\implies C=\left( \cfrac{18}{8}~~,~~\cfrac{21}{8} \right)
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C=\left( \cfrac{9}{4}~~,~~\cfrac{21}{8} \right)\implies C=\left(2\frac{1}{4}~~,~~2\frac{5}{8} \right)[/tex]
