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Question Number 208465 by Tawa11 last updated on 16/Jun/24

	Answered by A5T last updated on 17/Jun/24

	$\frac{3 \sqrt{2}}{s} = \frac{3 \sqrt{5}}{10} \Rightarrow s = \frac{10 \sqrt{10}}{5} = 2 \sqrt{10} \Rightarrow s^{2} = 40$
	Commented by Tawa11 last updated on 17/Jun/24

	$Thanks sir$
	Commented by Tawa11 last updated on 17/Jun/24

	$How do you get \frac{3 \sqrt{2}}{s} sir ?$
	Commented by A5T last updated on 17/Jun/24

	Commented by A5T last updated on 17/Jun/24

	$C o n s t r u c t s a y B D = B E = 3 \Rightarrow D E = 3 \sqrt{2},$ $a n d c o n s t r u c t D F = F G = E G = 3 \sqrt{2}$ $\Rightarrow B F = 3 \sqrt{5}; B H = r = 10$ $H o m o t h e t y t h a t m a p s F G t o H I g i v e s t h e$ $d e s i r e d s q u a r e a n d m a p s t h e c o n s t r u c t e d s q u a r e$ $a b o v e t o a n o t h e r s q u a r e w i t h t h e d e s i r e d$ $c o i n c i d e n t p r o p e r t i e s . T h i s h o m o t h e t y h a s r a t i o$ $\frac{B F}{B H} = \frac{3 \sqrt{5}}{r} = \frac{3 \sqrt{5}}{10} w h i c h i s a l s o e q u a l t o \frac{F G}{H I} = \frac{3 \sqrt{2}}{s}$ $\Rightarrow \frac{3 \sqrt{5}}{10} = \frac{3 \sqrt{2}}{s} \Rightarrow s = \frac{10 \sqrt{10}}{5} = 2 \sqrt{10}$
	Commented by A5T last updated on 17/Jun/24

	Commented by A5T last updated on 17/Jun/24

	$O n e c a n a l s o d e c i d e t o u s e a n y v a r i a b l e x .$ $C o n s t r u c t s a y B D = B E = x \Rightarrow D E = x \sqrt{2}$ $a n d c o n s t r u c t s a y D F = F G = E G = x \sqrt{2}$ $\Rightarrow B F = x \sqrt{5}; B H = r = 10$ $\frac{B F}{B H} = \frac{x \sqrt{5}}{10} = \frac{F G}{H I} = \frac{x \sqrt{2}}{s} \Rightarrow \frac{x \sqrt{5}}{10} = \frac{x \sqrt{2}}{s} \Rightarrow s = 2 \sqrt{10}$
	Commented by Tawa11 last updated on 17/Jun/24

	$Wow, I appreciate sir .$
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