Успоредник

[Гост]

Успоредник със страни а и b и диагонали d1 и d2 и ъгъл между тях [tex]\varphi[/tex]. Ако [tex]a^{2 }[/tex]+[tex]b^{2 }[/tex]=13 и d1 + d2 = 7 и [tex]\angle \varphi[/tex] = 45[tex]^\circ[/tex], колко е лицето на успоредника?

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[tex]\begin{array}{|l} d_1 + d_2 = 7 \Rightarrow d_2 = 7- d_1 \\ a^2 + b^2 = 13|.2 \Rightarrow 2(a^2 + b^2 ) = 26 \Rightarrow d_1 ^2 + d_2 ^2 = 26 \end{array}[/tex]

[tex]\begin{array}{|l} d_1 ^2 + d_2 ^2 = 26 \\ d_1 + d_2 = 7 \Rightarrow (d_1 + d_2 )^2 = 49 \Leftrightarrow d_1 ^2 + d_2 ^2 + 2 d_1 d_2 = 49 \Leftrightarrow 26 + 2 d_1 d_2 = 49 \end{array}[/tex]
[tex]d_1 d_2 = \frac {23}{2} \Rightarrow \frac {d_1 d_2}{4} = \frac {23}{8}[/tex]
[tex]S_ {ABCD} = 2 S _{\triangle AOB} + 2 S _ {\triangle AOD} = 2 (\frac 12 . \frac {d_1 d_2}{4} sin 135 ^\circ + \frac 12 . \frac {d_1 d_2}{4} sin 45 ^\circ)=[/tex]
[tex]= \frac {d_1 d_2}{4} (sin 135 ^\circ + sin 45 ^\circ) = \frac {23}{8} . \frac {2 \sqrt {2}}{2} = \frac {23 \sqrt {2}}{8}[/tex]

[Евва]

( 2 начин ) [tex]d_{1 } ^{2 }[/tex] +[tex]d_{2 } ^{2 }[/tex] =2([tex]a^{2 }[/tex] +[tex]b^{2 }[/tex]) ( формула)

[tex]( d_{1 } + d_{2 } )^{2 }[/tex] -2[tex]d_{1 }[/tex][tex]d_{2 }[/tex]=2.13

[tex]7^{2 }[/tex]-2[tex]d_{1 }[/tex][tex]d_{2 }[/tex]=26

[tex]d_{1 }[/tex][tex]d_{2 }[/tex]=[tex]\frac{23}{2}[/tex]

[tex]S_{ABCD }[/tex]=[tex]\frac{1}{2}[/tex].[tex]d_{1 }[/tex][tex]d_{2 }[/tex].sin45[tex]^\circ[/tex]

[tex]S_{ABCD }[/tex]=[tex]\frac{1}{2}[/tex].[tex]\frac{23}{2}[/tex].[tex]\frac{ \sqrt{2} }{2}[/tex] =[tex]\frac{23 \sqrt{2} }{8}[/tex]