[tex]\bf ~~~~~~~~~~~~\textit{internal division of a line segment}
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A(-9,5)\qquad B(15,-11)\qquad
\qquad \stackrel{\textit{ratio from A to B}}{7:1}
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\cfrac{A\underline{p}}{\underline{p} B} = \cfrac{7}{1}\implies \cfrac{A}{B} = \cfrac{7}{1}\implies 1A=7B\implies 1(-9,5)=7(15,-11)\\\\
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[tex]\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)\\\\
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p=\left(\cfrac{(1\cdot -9)+(7\cdot 15)}{7+1}\quad ,\quad \cfrac{(1\cdot 5)+(7\cdot -11)}{7+1}\right)
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p=\left(\cfrac{-9+105}{8}~~,~~\cfrac{5-77}{8} \right)\implies p=\left( \cfrac{96}{8}~~,~~\cfrac{-72}{8} \right)
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p=(12~,~-9)[/tex]
