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Question Number 36398 by NECx last updated on 01/Jun/18
$${Consider}\:{triangle}\:{ABC}.{If}\:\mathrm{206} \\ $$$${lines}\:{are}\:{drawn}\:{from}\:{A}\:{to}\:{BC}\:{how} \\ $$$${many}\:{triangles}\:{are}\:{formed}? \\ $$
Commented by Rasheed.Sindhi last updated on 02/Jun/18
$$\mathrm{207}+\mathrm{206}+...+\mathrm{1}=\frac{\mathrm{207}×\mathrm{208}}{\mathrm{2}}=\mathrm{21528} \\ $$$$\left(\mathrm{including}\:\bigtriangleup\mathrm{ABC}\right) \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 01/Jun/18
$${total}\:{lines}\mathrm{206}+{AB}+{AC}+{BC} \\ $$$${total}\:{points}\:{on}\:{BC}\:{for}\:{these}\:\mathrm{206}\:{lines}=\mathrm{206} \\ $$$${total}\:{points}\:{in}\:{BC}=\mathrm{2}{o}\mathrm{6}+{pointB}\:{and}\:{C} \\ $$$${any}\:{two}\:{points}\:{out}\:{of}\:{these}\mathrm{206}\:+\mathrm{2}=\mathrm{2}{o}\mathrm{8}\:{points} \\ $$$${when}\:{join}\:{withpoint}\:{A}...{make} \\ $$$${triangle}\: \\ $$$${so}\:{number}\:{of}\:\:{triangles}=\mathrm{2}{o}\mathrm{8}{C}_{\mathrm{2}} −\mathrm{1} \\ $$$${heree}\:\mathrm{1}\:{substructed}\:{for}\:{triangle}\:{ABC} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$