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Question Number 191756 by Shlock last updated on 30/Apr/23

	Answered by aleks041103 last updated on 30/Apr/23

	$s i d e o f w h o l e s q u a r e \to a$ $s i d e b l u e / r e d / y e l l o w s q u a r e \to b / r / y$ $b^{2} = r^{2} + {(a - 2 r)}^{2} = r^{2} + y^{2}$ $t h e r e a r e t w o s i m i l a r t r i a n g l e s$ $\frac{r}{y - r} = \frac{y}{r} \Rightarrow r^{2} = y^{2} - y r$ $\Rightarrow y^{2} - y r - r^{2} = 0$ $y + 2 r = a \Rightarrow y = a - 2 r > 0 \Rightarrow r < a / 2$ $\Rightarrow {(a - 2 r)}^{2} - (a - 2 r) r - r^{2} = 0$ $a^{2} - 4 a r + 4 r^{2} - a r + 2 r^{2} - r^{2} = 0$ $5 r^{2} - 5 a r + a^{2} = 0$ $r = \frac{5 \pm \sqrt{25 - 20}}{10} a = \frac{5 \pm \sqrt{5}}{10} a$ $r > 0 \Rightarrow r = \frac{5 - \sqrt{5}}{10} a \Rightarrow y = a (1 - \frac{5 - \sqrt{5}}{5}) = \frac{a}{\sqrt{5}} = y$ $a l s o$ $\frac{S_{y}}{S_{t o t}} = {(\frac{y}{a})}^{2} = \frac{1}{5} = \frac{S_{y}}{S_{t o t}}$
	Commented by Shlock last updated on 30/Apr/23
	Excellent!
	Answered by mr W last updated on 30/Apr/23
	$side of red square =a side of yellow square =b side of blue square =c=(√(a^2 +b^2 )) side of big white square =d=2a+b ((b−a)/a)=(a/b) ⇒((a/b))^2 +(a/b)−1=0 ⇒(a/b)=((−1+(√5))/2) fraction of yellow square: (b^2 /d^2 )=((b/(2a+b)))^2 =((1/(2×(a/b)+1)))^2 =((1/( (√5))))^2 =(1/5)$
	$s i d e o f r e d s q u a r e = a$ $s i d e o f y e l l o w s q u a r e = b$ $s i d e o f b l u e s q u a r e = c = \sqrt{a^{2} + b^{2}}$ $s i d e o f b i g w h i t e s q u a r e = d = 2 a + b$ $\frac{b - a}{a} = \frac{a}{b}$ $\Rightarrow {(\frac{a}{b})}^{2} + \frac{a}{b} - 1 = 0$ $\Rightarrow \frac{a}{b} = \frac{- 1 + \sqrt{5}}{2}$ $f r a c t i o n o f y e l l o w s q u a r e :$ $\frac{b^{2}}{d^{2}} = {(\frac{b}{2 a + b})}^{2} = {(\frac{1}{2 \times \frac{a}{b} + 1})}^{2} = {(\frac{1}{\sqrt{5}})}^{2} = \frac{1}{5}$
	Commented by Shlock last updated on 30/Apr/23
	Excellent!
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