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Question Number 190269 by normans last updated on 30/Mar/23

Commented by a.lgnaoui last updated on 02/Apr/23

	Answered by a.lgnaoui last updated on 02/Apr/23

	$T h e A r e a o f △ A E B A B C a n d △ A F C$ $a r e e q u a l t o y e l l o w △ A B C = 30 (A B = A C = B E = C F = a)$ $s u m o f A r e a = 30 \times 3 = 90$ $A r e a o f B C I ?$ $A B C (t r i a n g l e i s o c e l e a n d r e c t a n g l e) \Rightarrow$ $A r e a = \frac{a^{2}}{2} = 30 \Rightarrow a = 2 \sqrt{15} B C = A B \sqrt{2}$ $B C = 2 \sqrt{30} \Rightarrow A D = B D = \sqrt{30}$ $D I = A I - A D = 10 - \sqrt{30}$ $A r e a (△ B C I) = D I \times B D = B C \times D I = (10 - \sqrt{30}) \sqrt{30}$ $= 10 \sqrt{30} - 30$ $\Rightarrow A r e a o f g r e a t t r i a n g l e (E F I) i s$ $90 + 10 \sqrt{30} - 30 = 60 + 10 \sqrt{30}$ $t r i a n g l e s (E M I) (F N I) a r e e q u a l$ $t h e i r a r e a = A r e a t r a p e z e (E F M N) - (60 + 10 \sqrt{30})$ $A r e a (t r a p e z e) = (\frac{E F + M N}{2}) \times 10$ $= \frac{(2 B C + M N)}{2} \times 10 = 100 + 10 \sqrt{3}$ $A r e a (e q u a l t r i a n g l e s) = (100 + 10 \sqrt{3}) - (60 + 10 \sqrt{30}) = 40$ $\Rightarrow t h e A r e a o f f o r m s c o l o r e d i s$ $40 + a r e a (A B C I) = 40 + 10 \sqrt{30}$ $\Rightarrow T o t a l A r e a = 94, 77$
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