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Question Number 187167 by Rupesh123 last updated on 14/Feb/23

	Answered by mr W last updated on 14/Feb/23

	Commented by mr W last updated on 14/Feb/23

	$\sin θ = \frac{21}{49} = \frac{3}{7}$ $\cos 2 θ = 1 - 2 \times {(\frac{3}{7})}^{2} = \frac{31}{49}$ ${(2 x)}^{2} = 49^{2} + 85^{2} - 2 \times 49 \times 85 \times \frac{31}{49} = 4356$ $\Rightarrow x = \frac{\sqrt{4356}}{2} = 33 ✓$
	Commented by Rupesh123 last updated on 14/Feb/23
	Nice!
	Answered by mr W last updated on 14/Feb/23

	Commented by mr W last updated on 14/Feb/23

	$\cos θ = \frac{21}{49}$ $x^{2} = 21^{2} + 18^{2} + 2 \times 21 \times 18 \times \frac{21}{49} = 1089$ $\Rightarrow x = \sqrt{1089} = 33$
	Commented by Rupesh123 last updated on 15/Feb/23
	Good job!
	Answered by a.lgnaoui last updated on 14/Feb/23

	$t h e r e m o r e m e t h o d e s$ $S i m p l e m e t h o d e$ $△ A B C A N E (S e m b l a b l e s) B C / / D E$ $\frac{A B}{A N} = \frac{A C}{A E} = \frac{B C}{N E} = \frac{1}{2} \Rightarrow N E = 36$ $\Rightarrow B N = 21 A N = 42$ $△ A N E$ ${A N}^{2} = {A E}^{2} + {N E}^{2} - 2 A E \times N E \cos λ (1)$ $42^{2} = 36^{2} + 4 x^{2} - 4 x \times 36 \cos λ$ $= 144 x \cos λ = 4 x^{2} - 468$ $\cos λ = \frac{x^{2} - 117}{36 x} (1)$ $△ A D N A B e t p e r p e n d i c u l a i r e s$ ${B D}^{2} + {B N}^{2} = {D N}^{2}$ $D E = 85 = D N + N E = D N + 36 \Rightarrow D N = 49 (2)$ $D N = 49$ $△ A D E$ $(1) \Rightarrow {A D}^{2} = {D E}^{2} + {A E}^{2} - D E \times A E \cos λ$ $49^{2} = 85^{2} + 4 x^{2} - 4 \times 85 x \cos λ$ $49^{2} = 85^{2} + 4 x^{2} - 340 x \cos λ$ $340 x \cos λ = 4 x^{2} + 4824$ $\cos λ = \frac{x^{2} + 1206}{85 x} (2)$ $A B = B N \Rightarrow △ A D N : I s o c e l e (A D = D N)$ $△ A D E$ ${A D}^{2} = {A E}^{2} + {D E}^{2} - 2 A E \times D E \cos λ$ $49^{2} = 4 x^{2} + 85^{2} - 340 x \cos λ (2)$ $(1) e t (2) \Rightarrow \cos λ = \frac{x^{2} + 1206}{85 x} = \frac{x^{2} - 117}{36 x}$ $\frac{x^{2} + 1206}{85} = \frac{x^{2} - 117}{36}$ $85 (x^{2} - 117) = 36 (x^{2} + 1206)$ $x^{2} = \frac{85 \times 117 + 36 \times 1206}{85}$ $R e s u l t a t d e f i n i t i f :$ $x^{2} = \frac{9945 + 43416}{49} = 1089$ $x = 33$
	Commented by a.lgnaoui last updated on 14/Feb/23

	Commented by mr W last updated on 14/Feb/23

	$y e s, t h e r e a r e m a n y r i g h t m e t h o d s a n d$ $t h e r e a r e a l s o m a n y w r o n g m e t h o d s .$
	Commented by mr W last updated on 14/Feb/23

	$n o w y o u r a n s w e r i s c o r r e c t . w h e n y o u$ $t h i n k y o u r s o l u t i o n i s s i m p l e, {i t}^{'} s$ $y o u r r i g h t t o t h i n k w h a t y o u t h i n k .$
	Commented by a.lgnaoui last updated on 14/Feb/23

	$r e c t i f i c a t i o n c a l c u l$ $x^{2} = \frac{43416 + 9945}{49} = \frac{53361}{49}$ $x = 33$
	Commented by a.lgnaoui last updated on 14/Feb/23

	$i t s o n l y p r o b l e m e o p e r a t i o n$ $c a l c u l$
	Commented by a.lgnaoui last updated on 14/Feb/23

	$i t s a s i m p l e p r o b l e m e c a l c u l$ $s o r r y .$
	Commented by ajfour last updated on 14/Feb/23

	$d o n t u s e e t h e f o n t s i z e u u s e,$ $h o w d o y o u t y p e s o b i g ?$ $g u e s s i k n e w . .!$
	Commented by a.lgnaoui last updated on 14/Feb/23

	$y e s o k$
	Commented by Rupesh123 last updated on 15/Feb/23
	Good job!
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