Q214369


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Question Number 214623 by mr W last updated on 13/Dec/24

$Q 214369$
Commented by mr W last updated on 13/Dec/24

	Answered by mr W last updated on 13/Dec/24

	Commented by mr W last updated on 13/Dec/24

	$s = s i d e l e n g t h o f l a r g e e q u i l a t e r a l$ $t = s i d e l e n g t h o f s m a l l e q u i l a t e r a l (g r e e n)$ $P Q = s - 2 \sqrt{3} a$ $r e d l i n e = \sqrt{{(s - 2 \sqrt{3} a)}^{2} - {(2 a)}^{2}} = \sqrt{s^{2} + 8 a^{2} - 4 \sqrt{3} s a}$ $t = r e d l i n e$ $b = \frac{t}{2 \sqrt{3}} = \frac{1}{2} \sqrt{\frac{s^{2} + 8 a^{2} - 4 \sqrt{3} s a}{3}}$ $l e t α = \frac{a}{s}, β = \frac{b}{s}$ $\Rightarrow β = \frac{1}{2} \sqrt{\frac{1 + 8 α^{2} - 4 \sqrt{3} α}{3}}$ $s u c h t h a t b = a,$ $12 α^{2} = 1 + 8 α^{2} - 4 \sqrt{3} α$ $4 α^{2} + 4 \sqrt{3} α - 1 = 0$ $\Rightarrow α = 1 - \frac{\sqrt{3}}{2} \approx 0.134$
	Commented by ajfour last updated on 14/Dec/24

	$I h a d h o p e d i t w d b e d i f f i c u l t . Y o u$ $p r o v e d t h a t i t i s n t s o . c h e e r s .$
	Commented by ajfour last updated on 14/Dec/24
	https://youtu.be/_nOhjOVtvIQ?si=WlBbeWtccm0GgQxK
	Commented by ajfour last updated on 14/Dec/24

	Commented by ajfour last updated on 14/Dec/24

	$A b o v e y o u t u b e l i n k f o r v i d e o .$
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