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Question Number 205360 by Davidtim last updated on 18/Mar/24

	Answered by A5T last updated on 18/Mar/24

	Commented by A5T last updated on 19/Mar/24

	$F E = E G; E B = \sqrt{4^{2} + {(3 - F E)}^{2}}$ $∠ F E G = 180 - θ \Rightarrow ∠ F O G = θ (s i n c e O F E G i s c y c l i c)$ $\Rightarrow c o s θ = \frac{3 - F E}{\sqrt{25 - 6 F E + {F E}^{2}}}$ ${F G}^{2} = 2 {F E}^{2} (1 - c o s (180 - θ)) = 2 {F E}^{2} (1 + c o s θ)$ ${F G}^{2} = 1 + 1 - 2 c o s θ$ $\Rightarrow 2 - 2 c o s θ = 2 {F E}^{2} (1 + c o s θ) \Rightarrow 1 - c o s θ = {F E}^{2} + {F E}^{2} c o s θ$ $\Rightarrow 1 - {F E}^{2} = c o s θ (1 + {F E}^{2}) \Rightarrow c o s θ = \frac{1 - {F E}^{2}}{1 + {F E}^{2}}$ $\Rightarrow \frac{1 - {F E}^{2}}{1 + {F E}^{2}} = \frac{3 - F E}{\sqrt{25 - 6 F E + {F E}^{2}}} \Rightarrow F E = \frac{\sqrt{17} - 3}{2} = E G$ $A O = \sqrt{3^{2} + 1^{2}} = \sqrt{10} \land s i n D A O = \frac{\sqrt{10}}{10} = c o s (90 - D A O)$ ${O B}^{2} = {A O}^{2} + {A B}^{2} - 2 A O \times A B c o s (90 - D A O)$ $\Rightarrow {O B}^{2} = 10 + 16 - 2 \times 4 = 18 \Rightarrow O B = 3 \sqrt{2}$ $\Rightarrow G B = \sqrt{{O B}^{2} - {O G}^{2}} = \sqrt{18 - 1} = \sqrt{17}$ $\Rightarrow B E = E G + B G = \frac{3 \sqrt{17} - 3}{2} \Rightarrow {B E}^{2} = \frac{162 - 18 \sqrt{17}}{4}$ $\Rightarrow E C = \sqrt{{B E}^{2} - {B C}^{2}} = \sqrt{\frac{98 - 18 \sqrt{17}}{4}} = \frac{\sqrt{98 - 18 \sqrt{17}}}{2}$ $\Rightarrow Y e l l o w A r e a = \frac{E C \times B C}{2} = \sqrt{98 - 18 \sqrt{17}} s q . u n i t s$
	Commented by Davidtim last updated on 19/Mar/24

	${d i d n}^{'} t y o u f i n d a s h o r t m e t h o d w i t h o u t i t ?$
	Commented by A5T last updated on 19/Mar/24

	$N o, b u t t h e r e c o u l d b e . I s t h e a n s w e r c o r r e c t ?$
	Commented by mr W last updated on 19/Mar/24

	$\sqrt{98 - 18 \sqrt{17}} = 9 - \sqrt{17}$
	Commented by A5T last updated on 19/Mar/24

	$Y e a ✓$
	Answered by mr W last updated on 20/Mar/24

	Commented by mr W last updated on 19/Mar/24

	$y = \frac{x / 2}{4 - x / 2} \times 1 = \frac{x}{8 - x}$ $\frac{1}{2} \times 2 \times \frac{x}{2} = \frac{1}{2} \times (2 + \frac{x}{2} + \sqrt{2^{2} + {(\frac{x}{2})}^{2}}) \times (\frac{x}{8 - x})$ $6 - \frac{3 x}{2} = \sqrt{4 + \frac{x^{2}}{4}}$ $36 - 18 x + \frac{9 x^{2}}{4} = 4 + \frac{x^{2}}{4}$ $x^{2} - 9 x + 16 = 0$ $\Rightarrow x = \frac{9 - \sqrt{17}}{2}$ $y e l l o w a r e a = \frac{4 \times x}{2} = 9 - \sqrt{17}$
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