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Question Number 28501 by naka3546 last updated on 26/Jan/18

Commented by naka3546 last updated on 26/Jan/18

$$ABCDisnotsquare.$$$$LengthDE?$$

Answered by mrW2 last updated on 26/Jan/18

$$let\mathrm{\angle }ABD=\theta$$$$\mathrm{\angle }EBC=90-\theta$$$$\mathrm{\angle }BEC=\mathrm{\angle }BCE=\frac{1}{2}(180-\mathrm{\angle }EBC)=45+\frac{\theta }{2}$$$$\mathrm{\angle }ECF=90-(45+\frac{\theta }{2})=45-\frac{\theta }{2}$$$$AB=8+1=9$$$$CB=AD=9\tan \theta$$$$EC=8\cos (45-\frac{\theta }{2})$$$$EC=2\times 9\tan \theta \times \sin (45-\frac{\theta }{2})$$$$\Rightarrow 8\cos (45-\frac{\theta }{2})=2\times 9\tan \theta \times \sin (45-\frac{\theta }{2})$$$$\Rightarrow 4=9\tan \theta \times \tan (45-\frac{\theta }{2})$$$$\Rightarrow 4=9\tan \theta \times \frac{1-\tan \frac{\theta }{2}}{1+\tan \frac{\theta }{2}}$$$$\Rightarrow 4=9\times \frac{2\tan \frac{\theta }{2}}{1-{\tan }^2\frac{\theta }{2}}\times \frac{1-\tan \frac{\theta }{2}}{1+\tan \frac{\theta }{2}}$$$$\Rightarrow 2=9\times \frac{\tan \frac{\theta }{2}}{1+\tan \frac{\theta }{2}}\times \frac{1}{1+\tan \frac{\theta }{2}}$$$$\Rightarrow 2\times {(1+\tan \frac{\theta }{2})}^2=9\times \tan \frac{\theta }{2}$$$$\Rightarrow 4{\tan }^2\frac{\theta }{2}-5\tan \frac{\theta }{2}+2=0$$$$\Rightarrow (2\tan \frac{\theta }{2}-1)(\tan \frac{\theta }{2}-2)=0$$$$\Rightarrow \tan \frac{\theta }{2}=\frac{1}{2}or2\left(notsuitable\right)$$$$\Rightarrow \tan \theta =\frac{2\times \frac{1}{2}}{1-{\left(\frac{1}{2}\right)}^2}=\frac{4}{3}$$$$\Rightarrow \cos \theta =\frac{3}{5}$$$$EB=AD=9\tan \theta =9\times \frac{4}{3}=12$$$$DB=\frac{9}{\cos \theta }=\frac{9\times 5}{3}=15$$$$\Rightarrow DE=15-12=3$$

Answered by ajfour last updated on 26/Jan/18

$$letAD=BE=BC=y,DE=x,$$$$\mathrm{\angle }ABD=\theta ,then$$$$4\cos \theta =x\sin \theta ....\left(i\right)$$$$(x+y)\cos \theta =9....\left(ii\right)$$$$(x+y)\sin \theta =y....\left(iii\right)$$$$\Rightarrow \tan \theta =\frac{4}{x}=\frac{y}{9}\Rightarrow y=\frac{36}{x}$$$$also{(x+y)}^2=y^2+81$$$$\Rightarrow x^2+2xy=81$$$$x^2+2x\left(\frac{36}{x}\right)=81$$$$x^2=9orx=3.$$

Commented by mrW2 last updated on 27/Jan/18

$$\mathcal{G}reat!$$

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