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Question Number 180256 by mr W last updated on 09/Nov/22

Commented by HeferH last updated on 09/Nov/22

	Answered by HeferH last updated on 10/Nov/22

	$d + c = ?$ $∙ a + b = 8$ $∙ b + c = 11 (+)$ $∙ a + d = 9$ $a + b + d + c = 20$ $8 + d + c = 20$ $d + c = 12^{*}$
	Commented by mr W last updated on 10/Nov/22

	$r i g h t s i r! t h a n k s!$
	Answered by a.lgnaoui last updated on 10/Nov/22

	$A C ∣∣ E H E H A = 90$ $A i r e [A C E H] = (\frac{A C + E H}{2}) \times A H = a (\frac{2 c + a \sqrt{3}}{2}) = a c + \frac{a^{2}}{2} \sqrt{3}$ $A i r e [E H G] = \frac{a^{2} \sqrt{3}}{2}$ $A i r e t o t a l e = a c + a^{2} \sqrt{3} = 28 + x (x = A i r e (F I H G))$ $c a l u l d e x$ $x = H I \times a + F H \times I F (∡ F G H = 60) (1)$ $A i r e [A B H I] = 11 = \frac{A B + H I}{2} \times a$ $\tan 30 = \frac{H I}{a} H I = \frac{a}{3} \sqrt{3} x = \frac{2 a^{2}}{3} \sqrt{3} (2 \times A i r e [H I H]) (2)$ $(2) a c + a^{2} \sqrt{3} = 28 + 2 \frac{a^{2}}{3} \sqrt{3}$ $11 = \frac{(\frac{a \sqrt{3}}{3} + c)}{2} a \Rightarrow 22 = \frac{a^{2} \sqrt{3}}{3} + a c$ $a c = 22 - \frac{a^{2} \sqrt{3}}{3}$ $28 + x = (22 - \frac{a^{2} \sqrt{3}}{3}) + a^{2} \sqrt{3}$ $x = 2 \frac{a^{2} \sqrt{3}}{3} - 6 = \frac{a^{2} \sqrt{3}}{3}$ $\frac{a^{2} \sqrt{3}}{3} = 6 a^{2} = \frac{18 \sqrt{3}}{3} = 6 \sqrt{3}$ $D o n c A i r e = 6$
	Answered by mr W last updated on 10/Nov/22

	Commented by mr W last updated on 10/Nov/22

	$A_{1} = B_{1}$ $A_{2} = D_{2}$ $C_{2} = B_{2}$ $C_{1} = D_{1}$ $\Rightarrow A_{1} + A_{2} + C_{1} + C_{2} = B_{1} + B_{2} + D_{1} + D_{2}$ $\Rightarrow A + C = B + D$ $8 + x = 9 + 11$ $\Rightarrow x = 12 ✓$
	Commented by a.lgnaoui last updated on 10/Nov/22

	$t h a n k s$ $a p l i c a t i o n : t h e l o w o f m e d i a n e i n$ $t r i a n g l e A i r e (A 1) = A i r e (B 1) . . .$ $w i t h A i r e [A B I] = [A i r e (A 1)] + [A i r e (B 1)]$ $f i n a l y x = C 1 + C 2$
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