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Question Number 51558 by Tawa1 last updated on 28/Dec/18

Commented by Tawa1 last updated on 28/Dec/18

$$\mathrm{Please}\mathrm{help}.$$

Commented by Kunal12588 last updated on 28/Dec/18

Commented by Tawa1 last updated on 28/Dec/18

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

Answered by mr W last updated on 28/Dec/18

Commented by Tawa1 last updated on 28/Dec/18

$$\mathrm{I}\mathrm{reall}\mathrm{appreciate}\mathrm{your}\mathrm{time}\mathrm{sir}.\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

Commented by mr W last updated on 28/Dec/18

$$letAB=1$$$$AC=\frac{\sin 60°}{\sin 75°}=0.897$$$$BC=\frac{\sin 45°}{\sin 75°}=0.732$$$$ED=\frac{AB}{2}=0.5$$$$DF=\frac{AC}{2}=0.448$$$$FE=\frac{BC}{2}=0.366$$$$OE=OH=OG=OD=OI=OF=R$$$$=\frac{0.5\times 0.448\times 0.366}{\sqrt{(0.5+0.448+0.366)(-0.5+0.448+0.366)(0.5-0.448+0.366)(0.5+0.448-0.366)}}=0.259$$$$\sin \left(\mathrm{\angle }FOD/2\right)=\frac{FD}{2R}=\frac{0.448}{2\times 0.259}$$$$\Rightarrow \mathrm{\angle }FOF=119.734°$$$$\mathrm{\angle }FID=180°-119.734°/2=120.133°$$$$\mathrm{\angle }BDI=120.133-60=60.133°$$$$\mathrm{\angle }BID=180-120.133=59.867°$$$$DI=\frac{\sin 60°}{\sin 59.867°}\times \frac{BC}{2}=0.366$$$$BI=\frac{\sin 60.133°}{\sin 59.867°}\times \frac{BC}{2}=0.367$$$$FI=FB-BI=0.5-0.367=0.133$$$$\sin \frac{\mathrm{\angle }EOD}{2}=\frac{ED}{2R}=\frac{0.5}{2\times 0.259}$$$$\Rightarrow \mathrm{\angle }EOD=149.702°$$$$45+60-x/2=149/2$$$$\mathrm{\angle }HOG=2(45+60)-149.702=60.297°$$$$HG=2R\sin \frac{60.297°}{2}=0.260$$$$\mathrm{\angle }EOH=2\times 60-60.297=59.703°$$$$EH=2R\sin \frac{59.703°}{2}=0.258$$$$\mathrm{\angle }DOG=2\times 45-60.297=29.703°$$$$DG=2R\sin \frac{29.703°}{2}=0.133$$$$Areaofhexagon{A}_H$$$$A_H=\frac{1}{4}(0.366\times \sqrt{4\times 0.{259}^2-0.{366}^2}$$$$+0.258\times \sqrt{4\times 0.{259}^2-0.{258}^2}$$$$+0.260\times \sqrt{4\times 0.{259}^2-0.{260}^2}$$$$+0.133\times \sqrt{4\times 0.{259}^2-0.{133}^2}$$$$+0.366\times \sqrt{4\times 0.{259}^2-0.{366}^2}$$$$+0.133\times \sqrt{4\times 0.{259}^2-0.{133}^2})$$$$=\frac{1}{4}(0.134+0.116+0.116+0.067+0.134+0.067)$$$$=0.159$$$$Areaoftriangle{A}_T$$$$A_T=\frac{1}{2}\times 1\times 0.897\times \sin 45°=0.317$$$$\Rightarrow \frac{Areahexagon}{Areatriangle}=\frac{0.159}{0.317}=0.50$$

Commented by Tawa1 last updated on 28/Dec/18

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

Commented by Tawa1 last updated on 28/Dec/18

$$\mathrm{Is}\mathrm{that}\mathrm{the}\mathrm{final}\mathrm{answer}\mathrm{sir}$$

Commented by Tawa1 last updated on 28/Dec/18

$$\mathrm{When}\mathrm{you}\mathrm{are}\mathrm{free},\mathrm{help}\mathrm{me}\mathrm{finish}\mathrm{it}\mathrm{sir}.\mathrm{I}\mathrm{will}\mathrm{study}\mathrm{it}.\mathrm{Thanks}$$$$\mathrm{for}\mathrm{your}\mathrm{time}$$

Commented by peter frank last updated on 28/Dec/18

$$veryniceworksir.$$

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