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Question Number 206216 by cortano21 last updated on 09/Apr/24

	Answered by A5T last updated on 09/Apr/24

	$H C = \sqrt{108 + 4} = \sqrt{112} = 4 \sqrt{7}$ $L e t H C m e e t t h e c i r c u m c i r c l e o f t h e h e x a g o n$ $a t L t h e n, H C \times H L = F H \times H E = 8 \Rightarrow H L = \frac{2 \sqrt{7}}{7}$ $s i n ∠ E H C = \frac{\sqrt{108}}{\sqrt{112}} = \frac{3 \sqrt{21}}{14} \Rightarrow c o s ∠ E H C = \frac{\sqrt{7}}{14}$ $\Rightarrow \sqrt{x^{2} + 2^{2} - 4 x \frac{\sqrt{7}}{14}} = \sqrt{{K D}^{2} + 6^{2} - 6 \sqrt{3} \times K D}$ $\Rightarrow x^{2} + 4 - \frac{2 x \sqrt{7}}{7} = {K D}^{2} + 36 - 6 \sqrt{3} K D . . . (i)$ $F K \times K D = K C \times K L$ $\Rightarrow (\sqrt{108} - K D) K D = (4 \sqrt{7} - x) (x + \frac{2 \sqrt{7}}{7})$ $\Rightarrow 6 \sqrt{3} K D - {K D}^{2} = 4 x \sqrt{7} + 8 - x^{2} - \frac{2 x \sqrt{7}}{7} . . . (i i)$ $(i) & (i i) \Rightarrow 32 + \frac{2 x \sqrt{7}}{7} = 4 x \sqrt{7} + 8 - \frac{2 x \sqrt{7}}{7}$ $\Rightarrow x = \sqrt{7} \approx 2.64575$
	Answered by mr W last updated on 09/Apr/24

	Commented by mr W last updated on 09/Apr/24

	$H C = \sqrt{12^{2} + 8^{2} - 12 \times 8} = 4 \sqrt{7}$ $\frac{x}{y} \times \frac{6}{6} \times \frac{12}{4} = 1$ $\Rightarrow \frac{x}{y} = \frac{1}{3} \Rightarrow \frac{x}{x + y} = \frac{1}{1 + 3} = \frac{1}{4}$ $\Rightarrow x = \frac{H C}{4} = \sqrt{7}$
	Answered by cortano21 last updated on 10/Apr/24

	Commented by mr W last updated on 10/Apr/24

	$F K = \frac{F D}{2} i s n o t o b v i o u s, m u s t b e$ $p r o v e d a t f i r s t .$
	Commented by A5T last updated on 10/Apr/24

	$\frac{4}{8} \times \frac{12}{6} \times \frac{D K}{K F} = 1 \Rightarrow D K = K F \Rightarrow F K = \frac{F D}{2}$
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