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Question Number 54206 by ajfour last updated on 31/Jan/19

Commented by ajfour last updated on 31/Jan/19

$F i n d t h e a r e a o f t r a p e z i u m A B C D$ $i n t e r m s o f a . (t h e r e d c i r c l e s a r e e a c h$ $w i t h i n a h a l f q u a r t e r c i r c l e a n d t h e$ $o u t e r b o u n d a r y i s a s q u a r e o f s i d e a) .$
Commented by ajfour last updated on 31/Jan/19

$(l i k e c h i l d r e n, w e s h o u l d s t i l l p l a y!)$
	Answered by mr W last updated on 31/Jan/19

	$\frac{c}{\tan \frac{45 °}{2}} + \sqrt{{(a + c)}^{2} - c^{2}} = \sqrt{2} a$ $\Rightarrow (\sqrt{2} + 1) c + \sqrt{a (a + 2 c)} = \sqrt{2} a$ $a (a + 2 c) = 2 a^{2} + (3 + 2 \sqrt{2}) c^{2} - 2 (\sqrt{2} + 1) c \sqrt{a (a + 2 c)}$ $a^{2} + (3 + 2 \sqrt{2}) c^{2} - 2 a c = 2 (\sqrt{2} + 1) c \sqrt{a (a + 2 c)}$ ${[a^{2} + (3 + 2 \sqrt{2}) c^{2} - 2 a c]}^{2} = 4 (3 + 2 \sqrt{2}) a (a + 2 c) c^{2}$ $a^{4} + (17 + 12 \sqrt{2}) c^{4} + 4 a^{2} c^{2} + 2 (3 + 2 \sqrt{2}) a^{2} c^{2} - 4 a^{3} c - 4 (3 + 2 \sqrt{2}) {a c}^{3} = 4 (3 + 2 \sqrt{2}) (a^{2} c^{2} + 2 {a c}^{3})$ $\Rightarrow (17 + 12 \sqrt{2}) c^{4} - 6 (6 + 4 \sqrt{2}) {a c}^{3} + 4 a^{2} c^{2} - 2 (3 + 2 \sqrt{2}) a^{2} c^{2} - 4 a^{3} c + a^{4} = 0$ $\Rightarrow c = . . . i n t e r m s o f a$ $(a - b) \sin \frac{45 °}{2} = b$ $(a - b) \frac{\sqrt{2 - \sqrt{2}}}{2} = b$ $\Rightarrow b = \frac{\sqrt{2 - \sqrt{2}} a}{2 + \sqrt{2 - \sqrt{2}}}$ $. . . .$
	Commented by ajfour last updated on 31/Jan/19

	$(\sqrt{2} + 1) c + \sqrt{a (a + 2 c)} = \sqrt{2} a$ $l e t a + 2 c = {a t}^{2}$ $\Rightarrow a (\sqrt{2} + 1) (t^{2} - 1) + 2 a t = 2 \sqrt{2} a$ $\Rightarrow (\sqrt{2} + 1) t^{2} + 2 t - 2 \sqrt{2} - (\sqrt{2} + 1) = 0$ $\Rightarrow t = \frac{- 1 + \sqrt{1 + 2 \sqrt{2} (\sqrt{2} + 1) + {(\sqrt{2} + 1)}^{2}}}{(\sqrt{2} + 1)}$ $= \frac{- 1 + \sqrt{1 + 4 + 2 \sqrt{2} + 3 + 2 \sqrt{2}}}{\sqrt{2} + 1}$ $t = \frac{- 1 + 2 \sqrt{2 + \sqrt{2}}}{\sqrt{2} + 1}$ $\Rightarrow t^{2} - 1 = \frac{1 + 8 + 4 \sqrt{2} - 4 \sqrt{2 + \sqrt{2}}}{3 + 2 \sqrt{2}} - 1$ $= \frac{6 + 2 \sqrt{2} - 4 \sqrt{2 + \sqrt{2}}}{3 + 2 \sqrt{2}}$ $c = \frac{a}{2} (t^{2} - 1) = a (\frac{3 + \sqrt{2} - 2 \sqrt{2 + \sqrt{2}}}{3 + 2 \sqrt{2}})$ $(c \approx 0.12331 a)$ $. . . . p l e a s e c h e c k S i r, b = \frac{a \sqrt{2 - \sqrt{2}}}{2 + \sqrt{2 - \sqrt{2}}}$ $o r b = \frac{\sqrt{2} a}{2 \sqrt{2 + \sqrt{2}} + \sqrt{2}} .$
	Commented by mr W last updated on 31/Jan/19

	$v e r y v e r y f i n e s i r!$
	Commented by ajfour last updated on 31/Jan/19

	$t h a n k s, n o t m u c h r e m a i n s t o b e$ $d o n e, S i r .$ $A = (b + c) [a \sqrt{2} - (\sqrt{2} + 1) (b + c)] .$
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