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Question Number 195750 by sonukgindia last updated on 09/Aug/23

Commented by mr W last updated on 09/Aug/23

$$?=3\times (8+8)-8-15=25$$

Commented by deleteduser4 last updated on 10/Aug/23

$$pleaseexplainyoursolution$$$$thankyou$$

Answered by mr W last updated on 10/Aug/23

Commented by mr W last updated on 10/Aug/23

$$d=\frac{a+b}{2}$$$$b=\frac{c+d}{2}\Rightarrow c=2b-d=2b-\frac{a+b}{2}=\frac{3b-a}{2}$$$$BE=\sqrt{3}b$$$$AD=\sqrt{3}d$$$$\frac{\sqrt{3}ab}{2}=2\times 8$$$$\Rightarrow ab=\frac{32}{\sqrt{3}}$$$$\frac{\sqrt{3}cd}{2}=2\times 15$$$$\Rightarrow cd=\frac{60}{\sqrt{3}}$$$$\left(\frac{3b-a}{2}\right)\left(\frac{a+b}{2}\right)=\frac{60}{\sqrt{3}}$$$$(3\sqrt{3}{b}^2-\sqrt{3}ab)(\sqrt{3}ab+\sqrt{3}{b}^2)=240\sqrt{3}{b}^2$$$$(3\sqrt{3}{b}^2-32)(32+\sqrt{3}{b}^2)=240\sqrt{3}{b}^2$$$$lett=\sqrt{3}{b}^2$$$$(3t-32)(t+32)=240t$$$$3t^2-176t-1024=0$$$$\Rightarrow t=64=\sqrt{3}{b}^2$$$$\Rightarrow b=\frac{8}{\sqrt[4]{3}}$$$$\Rightarrow a=\frac{32}{\sqrt{3}b}=\frac{4}{\sqrt[4]{3}}$$$$\Rightarrow \frac{b}{a}=2$$$${\mathrm{\Delta }}_{BCE}=2{\mathrm{\Delta }}_{ABE}$$$$\Rightarrow {\mathrm{\Delta }}_{ACE}=3{\mathrm{\Delta }}_{ABE}=3\times (8+8)=48$$$$A_{red}=48-8-15=25✓$$

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