Question Number 42711 by rahul 19 last updated on 01/Sep/18
$$\mathrm{If}\:\mathrm{2}\:\mathrm{sides}\:\mathrm{of}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{are}\:\hat {\mathrm{i}}+\mathrm{2}\hat {\mathrm{j}}\:\mathrm{and} \\ $$$$\hat {\mathrm{i}}+\hat {\mathrm{k}}\:,\:\mathrm{then}\:\mathrm{find}\:\mathrm{all}\:\mathrm{possible}\:\mathrm{third}\:\mathrm{side}\:? \\ $$
Answered by MJS last updated on 01/Sep/18
$$\mathrm{any}\:\mathrm{triangle}: \\ $$$${a}+{b}>{c} \\ $$$${a}+{c}>{b} \\ $$$${b}+{c}>{a} \\ $$$$ \\ $$$$\mathrm{this}\:\mathrm{case}: \\ $$$${i}+\mathrm{2}{j}+{i}+{k}>{c}\:\Rightarrow\:{c}<\mathrm{2}{i}+\mathrm{2}{j}+{k} \\ $$$${i}+\mathrm{2}{j}+{c}>{i}+{k}\:\Rightarrow\:{c}>{k}−\mathrm{2}{j} \\ $$$${i}+{k}+{c}>{i}+\mathrm{2}{j}\:\Rightarrow\:{c}>\mathrm{2}{j}−{k} \\ $$$$ \\ $$$$\Rightarrow\:\mathrm{case}\:\mathrm{1}:\:{k}−\mathrm{2}{j}<{c}<\mathrm{2}{i}+\mathrm{2}{j}+{k}\:\wedge\:\mathrm{2}{j}<{k} \\ $$$$\:\:\:\:\:\:\mathrm{case}\:\mathrm{2}:\:\mathrm{2}{j}−{k}<{c}<\mathrm{2}{i}+\mathrm{2}{j}+{k}\:\wedge\:{k}<\mathrm{2}{j} \\ $$
Commented by rahul 19 last updated on 02/Sep/18
$$\mathrm{ok}\:\mathrm{sir}\:\mathrm{thnks}. \\ $$