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Question Number 47282 by behi83417@gmail.com last updated on 07/Nov/18

Answered by MrW3 last updated on 07/Nov/18

Commented by MrW3 last updated on 08/Nov/18

$$AB=c$$$$AD=DB=\frac{c}{2}$$$$CD=m_c$$$$let\lambda =\frac{c}{2m_c}$$$$\frac{m_c}{\sin A}=\frac{c}{2\sin \alpha }$$$$\Rightarrow \sin \alpha =\frac{c}{2m_c}\sin A=\lambda \sin A$$$$similarly$$$$\sin \beta =\lambda \sin B$$$$\alpha +\beta =\mathrm{\angle }C=\pi -(\mathrm{\angle }A+\mathrm{\angle }B)$$$$\Rightarrow \sin (\alpha +\beta )=\sin (A+B)$$$$\Rightarrow \sin \alpha \cos \beta +\cos \alpha \sin \beta =\sin (A+B)$$$$\Rightarrow \lambda (\sin A\sqrt{1-{\lambda }^2{\sin }^{2}B}+\sin B\sqrt{1-{\lambda }^2{\sin }^{2}A})=\sin (A+B)$$$$A=20°,B=45°$$$$A+B=65°\ne 90°$$$$\Rightarrow \lambda =1(notsuitablesinceA+B\ne 90°)$$$$\Rightarrow \lambda =1.4112$$$$\Rightarrow c=2\times 1.4112\times \sqrt{5}=6.3111$$$$\frac{h}{\tan A}+\frac{h}{\tan B}=c$$$$\Rightarrow h=\frac{c}{\cot A+\cot B}=1.6841$$$$A_{\mathrm{\Delta }ABC}=\frac{1}{2}ch=\frac{1}{2}\times 6.3111\times 1.6841$$$$=5.314.$$

Commented by behi83417@gmail.com last updated on 07/Nov/18

$$thanksinadvancedearmaster.$$

Answered by behi83417@gmail.com last updated on 07/Nov/18

Commented by MrW3 last updated on 08/Nov/18

$$nicemethodsir!$$

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