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Question Number 162598 by amin96 last updated on 30/Dec/21

Answered by mr W last updated on 30/Dec/21

Commented by mr W last updated on 30/Dec/21

say side length of hexagon is 1.  then the side length of square is 2.  AK=2−1×cos 30°=2−((√3)/2)  tan α=((KI)/(AK))=(1/(2(2−((√3)/2))))=(1/(4−(√3)))  LE=((AL)/(tan α))=4−(√3)  ME=LE−2=2−(√3)  tan (x/2)=((BM)/(ME))=(1/(2−(√3)))=2+(√3)  tan x=((2(2+(√3)))/(1−(2+(√3))^2 ))=−(1/( (√3)))  ⇒x=180°−tan^(−1) (1/( (√3)))=180°−30°=150°

$${say}\:{side}\:{length}\:{of}\:{hexagon}\:{is}\:\mathrm{1}. \\ $$$${then}\:{the}\:{side}\:{length}\:{of}\:{square}\:{is}\:\mathrm{2}. \\ $$$${AK}=\mathrm{2}−\mathrm{1}×\mathrm{cos}\:\mathrm{30}°=\mathrm{2}−\frac{\sqrt{\mathrm{3}}}{\mathrm{2}} \\ $$$$\mathrm{tan}\:\alpha=\frac{{KI}}{{AK}}=\frac{\mathrm{1}}{\mathrm{2}\left(\mathrm{2}−\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\right)}=\frac{\mathrm{1}}{\mathrm{4}−\sqrt{\mathrm{3}}} \\ $$$${LE}=\frac{{AL}}{\mathrm{tan}\:\alpha}=\mathrm{4}−\sqrt{\mathrm{3}} \\ $$$${ME}={LE}−\mathrm{2}=\mathrm{2}−\sqrt{\mathrm{3}} \\ $$$$\mathrm{tan}\:\frac{{x}}{\mathrm{2}}=\frac{{BM}}{{ME}}=\frac{\mathrm{1}}{\mathrm{2}−\sqrt{\mathrm{3}}}=\mathrm{2}+\sqrt{\mathrm{3}} \\ $$$$\mathrm{tan}\:{x}=\frac{\mathrm{2}\left(\mathrm{2}+\sqrt{\mathrm{3}}\right)}{\mathrm{1}−\left(\mathrm{2}+\sqrt{\mathrm{3}}\right)^{\mathrm{2}} }=−\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}} \\ $$$$\Rightarrow{x}=\mathrm{180}°−\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}=\mathrm{180}°−\mathrm{30}°=\mathrm{150}° \\ $$

Commented by amin96 last updated on 30/Dec/21

thanks sir. greatful

$${thanks}\:{sir}.\:{greatful} \\ $$

Commented by Tawa11 last updated on 30/Dec/21

Great sir. God bless you

$$\mathrm{Great}\:\mathrm{sir}.\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you} \\ $$

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