TEOREMA DEI SENI

a = 12√2 = 16,97..

α = 180-(60+45) = 75°

α/sin 75 = b/sin 60

75° = 45°+30°

sin (α+β) = sin α*cos β + cos α*sin β

sin 75° = √2 /2 * √3 /2 + √2 /2 * 0,5 = √6 /4+√2 /4 = (√6+√2)/4

12√2 / (√6+√2)/4  = b / sin 60°

b = 48√2*√3 /2 / (√6+√2) =  24√6 / (√6+√2) = 24√6 * (√6-√2) /4

b = (24*6-24√12)/4 = 24(6-2√3))/4 = 48*(3-√3)/4 = 12(3-√3) ..15,22

b/sin β = c/sin ϒ

c = b*sin ϒ /sin β

c = b*√2 /√3   (...12,42) 

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Angolo $\small \alpha= 180°-\beta-\gamma = 180-60-45 = 75°;$

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$\small b= \dfrac{a·\sin(\beta)}{\sin(\alpha)}= \dfrac{12\sqrt2·\sin(60°)}{\sin(75°)}$

$\small b= \dfrac{\cancel{12}^6\sqrt2·\dfrac{\sqrt3}{\cancel2_1}}{\dfrac{\sqrt6 +\sqrt2}{4}}$

$\small b= 6\sqrt2·\sqrt3·\dfrac{4}{\sqrt6+\sqrt2}$

$\small b= \dfrac{24\sqrt6}{\sqrt6+\sqrt2}$

$\small b= \dfrac{24\sqrt6}{\sqrt6+\sqrt2}·\dfrac{\sqrt6-\sqrt2}{\sqrt6-\sqrt2}$

$\small b= \dfrac{24\sqrt6·\sqrt6-24\sqrt6·\sqrt2}{(\sqrt6)^2-(\sqrt2)^2}$

$\small b= \dfrac{24·6-24\sqrt{12}}{6-2}$

$\small b= \dfrac{144-24·2\sqrt3}{4}$

$\small b= \dfrac{144-48\sqrt3}{4}$

$\small b= \dfrac{\cancel{48}^{12}(3-\sqrt3)}{\cancel4_1}$

$\small b= 12(3-\sqrt3) \approx{15,2154};$

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$\small c= \dfrac{a·\sin(\gamma)}{\sin(\alpha)} = \dfrac{12\sqrt2·\sin(45°)}{\sin(75°)}$

$\small c= \dfrac{\cancel{12}^6\sqrt2·\dfrac{\sqrt2}{\cancel2_1}}{\dfrac{\sqrt6+\sqrt2}{4}}$

$\small c= 6·2·\dfrac{4}{\sqrt6+\sqrt2}$

$\small c= \dfrac{48}{\sqrt6+\sqrt2}$

$\small c= \dfrac{48}{\sqrt6+\sqrt2}·\dfrac{\sqrt6-\sqrt2}{\sqrt6-\sqrt2}$

$\small c= \dfrac{48(\sqrt6-\sqrt2)}{(\sqrt6)^2-(\sqrt2)^2}$

$\small c= \dfrac{48(\sqrt6-\sqrt2)}{6-2}$

$\small c= \dfrac{\cancel{48}^{12}(\sqrt6-\sqrt2)}{\cancel4_1}$

$\small c= 12(\sqrt6-\sqrt2) \approx{12,4233}.$