Verified answer

Угол CAB = 180° - 135° = 45°

Угол ACB = 180° - (45° + 75°) = 180° - 120° = 60°

   Для удобства обозначим:

CAB = α

ABC = β

ACB = γ

   По теореме синусов:

[tex]\displaystyle AB = \frac{BC\sin\gamma}{sin\alpha}[/tex]

[tex]\displaystyle AC = \frac{BC\sin\beta}{sin\alpha}[/tex]

   Площадь можно найти по формуле:

[tex]\displaystyle S = \frac{1}{2}AB*AC*\sin\alpha[/tex]

   В итоге:

[tex]\displaystyle S = \frac{BC^2 \sin\beta sin\gamma }{2\sin\alpha }[/tex]

[tex]\displaystyle S = \frac{12^2 \sin75^{\circ} sin60^{\circ} }{2\sin 45^{\circ}}[/tex]

[tex]\displaystyle S = \frac{144*\sin(30^{\circ} + 45^{\circ})\sin60^{\circ}}{2\sin45^{\circ}}[/tex]

[tex]\displaystyle S = \frac{144 * \sin(30^{\circ} + 45^{\circ} )\frac{\sqrt{3} }{2} }{2\frac{\sqrt{2} }{2} }[/tex]

[tex]\displaystyle S = 72\sqrt{\frac{3}{2} }(\sin30^{\circ}\cos45^{\circ}+\cos30^{\circ}\sin45^{\circ})[/tex]

[tex]\displaystyle S = 72\sqrt{\frac{3}{2} }(\frac{\sqrt{2} + \sqrt{6} }{4} )[/tex]

[tex]\displaystyle S = 18\sqrt{3}(1+\sqrt{3} ) = 18\sqrt{3} + 54[/tex]