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Question Number 5825 by Rasheed Soomro last updated on 30/May/16

Commented by Rasheed Soomro last updated on 30/May/16

$$\mathrm{A}\mathrm{is}\mathrm{the}\mathrm{center}\mathrm{of}\mathrm{arc}\mathrm{BC}.$$$$\mathrm{AB}=\mathrm{AE}=\mathrm{AG}=\mathrm{AC}$$$$\mathrm{AB}m\mathrm{eans}\mathrm{distance}\mathrm{between}\mathrm{A}\mathrm{and}\mathrm{B}$$$$\mathrm{Similarly}\mathrm{AE},\mathrm{AG},\mathrm{AC}\mathrm{are}\mathrm{also}\mathrm{distances}.$$

Answered by Yozzii last updated on 30/May/16

$$ThecentreofthearcAEisBand$$$$\mid AB\mid =\mid AE\mid >0.SinceAEisanarc$$$$\Rightarrow \mid AB\mid =\mid BE\mid and\mid AE\mid =\mid AB\mid \theta$$$$where\theta =\mathrm{\angle }ABE.\therefore \theta =\frac{\mid AE\mid }{\mid AB\mid }=1rad.$$$$CisthecentreofthearcAGand$$$$\mid AG\mid =\mid AC\mid .Wesimilarlyfind\mathrm{\angle }ACG=\alpha =1rad.$$$$WearegiventhatA,B\mathrm{\&}Care$$$$collinearpoints.So\mathrm{\angle }BAC=\gamma =\pi rad.$$$$AisthecentreofthearcBC.$$$$Wenowemploytheformulafor$$$$theareaofthesectorswhoseangles$$$$are\theta ,\alpha and\gamma respectively.$$$$A_{\theta }=\frac{1}{2}{(\mid AB\mid )}^2\theta =\frac{{\left(AB\right)}^2}{2}$$$$A_{\alpha }=\frac{1}{2}{(\mid AC\mid )}^2\alpha =\frac{{(\mid AC\mid )}^2}{2}=\frac{{(\mid AB\mid )}^2}{2}\because \mid AB\mid =\mid AC\mid$$$$A_{\gamma }=\frac{1}{2}{(\mid AB\mid )}^2\gamma =\frac{{(\mid AB\mid )}^2}{2}\pi .$$$$Thetotalareaoftheclosedfigure$$$$isthenA=A_{\gamma }+A_{\theta }+A_{\alpha }or$$$$A=\frac{{(\mid AB\mid )}^2}{2}(2+\pi )=\frac{r^2}{2}(2+\pi )\because \mid AB\mid =r.$$

Commented by Rasheed Soomro last updated on 30/May/16

$$\mathrm{Very}\mathrm{sorry}\mathrm{that}\mathrm{my}\mathrm{question}\mathrm{was}\mathrm{not}\mathrm{clear}.$$$$\mathrm{According}\mathrm{to}\mathrm{your}\mathrm{understanding}\mathrm{you}\mathrm{are}$$$$\mathrm{very}\mathrm{right}!$$$$\mathrm{But}\mathrm{I}{\mathrm{didn}}^{\prime }\mathrm{t}\mathrm{mean}\mathrm{that}.$$$$\mathrm{By}\mathrm{AE},\mathrm{AB},\mathrm{AG},...\mathrm{I}\mathrm{meant}\mathrm{distances}\mathrm{between}$$$$\mathrm{points}.\mathrm{Where}\mathrm{I}\mathrm{meant}\mathrm{arc}\mathrm{I}\mathrm{wrote}\mathrm{arc}\mathrm{AG}etc.$$

Commented by Yozzii last updated on 30/May/16

$$Mycalculationconsidersthesymbolism$$$$youusedinitiallyasmeasuresof$$$$distancesbetweenpoints.If{you}^{\prime }d$$$$like,wecanwriteforexamplethedistance$$$$betweenpointsAandBas\mid AB\mid .$$$$$$

Commented by Rasheed Soomro last updated on 07/Jun/16

$$\underleftarrow{!\mathcal{S}kna\mathcal{HT}}$$

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