Question Number 198161 by Blackpanther last updated on 12/Oct/23
Answered by Rasheed.Sindhi last updated on 12/Oct/23
$$\blacktriangle\mathrm{ABC}=\frac{\mathrm{1}}{\mathrm{2}}.\mathrm{AC}.\mathrm{BC} \\ $$$$\:\:\:\:\:\:\:=\frac{\mathrm{1}}{\mathrm{2}}×\mathrm{12}×{y}=\mathrm{6}{y} \\ $$$$\:\:\:\:\:\:\:\mathrm{48}\leqslant\mathrm{6}{y}\leqslant\mathrm{60} \\ $$$$\:\:\:\:\:\:\:\mathrm{8}\leqslant{y}\leqslant\mathrm{10} \\ $$$$\because\:{y}\in\mathbb{Z} \\ $$$$\therefore\:{y}=\left\{\mathrm{8},\mathrm{9},\mathrm{10}\right\} \\ $$$${Now}, \\ $$$$\mathrm{tan}\:\mathrm{A}=\frac{{y}}{\mathrm{12}}=\frac{{x}}{\mathrm{5}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{12}{x}=\mathrm{5}{y} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}=\frac{\mathrm{5}{y}}{\mathrm{12}} \\ $$$$\because\:{y}=\mathrm{8},\mathrm{9},\mathrm{10} \\ $$$$\therefore\:{x}=\frac{\mathrm{5}\centerdot\mathrm{8}}{\mathrm{12}},\:\frac{\mathrm{5}\centerdot\mathrm{9}}{\mathrm{12}},\:\frac{\mathrm{5}\centerdot\mathrm{10}}{\mathrm{12}} \\ $$$$\:\:\:\:\:\:\:\:=\frac{\mathrm{10}}{\mathrm{3}},\:\frac{\mathrm{15}}{\mathrm{4}},\:\frac{\mathrm{25}}{\mathrm{6}} \\ $$