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Question Number 186402 by Rupesh123 last updated on 04/Feb/23

Answered by mr W last updated on 04/Feb/23

Commented by mr W last updated on 04/Feb/23

$$\sin \alpha =\frac{4\sin 75°}{6}$$$$\Rightarrow \frac{\sin 75°}{\sin \alpha }=\frac{6}{4}=\frac{3}{2}$$$$\theta =105°-\alpha$$$$\gamma =\theta -30°=75°-\alpha$$$$\frac{x}{\sin \alpha }=\frac{4+6}{\sin \theta }$$$$\Rightarrow x=\frac{10\sin \alpha }{\sin \theta }$$$$\frac{y}{\sin 30°}=\frac{x}{\sin \gamma }$$$$\Rightarrow y=\frac{x}{2\sin \gamma }$$$$greenarea=\frac{xy\sin \theta }{2}$$$$=\frac{25{\sin }^2\alpha }{\cos (\alpha +15°)\cos (\alpha -15°)}$$$$=\frac{25{\sin }^2\alpha }{{\cos }^{2}15-{\sin }^2\alpha }$$$$=\frac{25}{{\left(\frac{\sin 75°}{\sin \alpha }\right)}^2-1}$$$$=\frac{25}{{\left(\frac{3}{2}\right)}^2-1}=20✓$$

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