Geometry

$\gamma =atan \frac{4}{3}\\ \gamma =53.1301^{\circ}\\ \alpha=\arcsin \frac{3}{5}\\ \alpha =33.8699^{\circ}\\ \color{blue}\frac{\alpha}{2}=18.4349^{\circ}$

$\angle ADC=(180-53.1301-18.4349)^{\circ}\\ \angle ADC=108.4349^{\circ}$

$tan \frac{\\alpha}{2}=\frac {\overline{BD}}{4}\\ \overline{BD}=4\cdot tan\ 18.4349^{\circ}\\ \overline{BD}=1\frac{1}{3}\\ \overline{DC}=3-1\frac{1}{3}\\ \overline{DC}=\frac{5}{3}$

$A_{ADC}=\frac{\overline{DC} \cdot \overline{AB}}{2}=\frac{5}{3}\cdot \frac{4}{2}=\frac{20}{6}\\ \color{blue}A_{ADC}=3\frac{1}{3}$

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