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Question Number 35798 by ajfour last updated on 23/May/18

Commented by ajfour last updated on 23/May/18

$$△ABCand△CDEarecongruent.$$$$Findtheareaofquadrilateral$$$$APQCintermsofa,b,andc;the$$$$sidesof△ABC.$$

Commented by ajfour last updated on 23/May/18

Commented by ajfour last updated on 23/May/18

$$considering△ADP$$$$\frac{AP}{\sin \mathrm{\angle }ADP}=\frac{AD}{\sin \mathrm{\angle }APD}$$$$\Rightarrow \frac{c-BP}{\sin B}=\frac{c-b}{\sin (\theta +B)}$$$$\theta =\pi -\mathrm{\angle }A$$$$soBP=c-\frac{(c-b)\sin B}{\sin (A-B)}$$$$.......................$$$$considering△CEQ$$$$\frac{CQ}{\sin \mathrm{\angle }CEQ}=\frac{CE}{\sin \mathrm{\angle }CQE}$$$$\Rightarrow \frac{a-BQ}{\sin C}=\frac{b}{\sin \theta (=\sin A)}$$$$BQ=a-\frac{b\sin C}{\sin A}$$$$.........................$$$$Area(△PBQ)=\frac{BP\times BQ\sin B}{2}$$$$Area\left(APQC\right)=Area(△ABC)$$$$-Area(△PBQ).$$

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