Find the angle $ \alpha $ of a rhombus if side $a = \frac{ 353 }{ 25 }$ and diagonal $d_2 = 11$.

$\alpha = 50.0383^o$

STEP 1: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

$$ d_1^2 + d_2^2 = 4 \cdot a^2 $$

After substituting $ d_2 = 11 $ and $ a = \frac{ 353 }{ 25 } $ we have:

$$ d_1 ^ {\,2} + 11^2 = 4 \cdot \left(\frac{ 353 }{ 25 }\right)^2 $$ $$ d_1 ^ {\,2} + 121 = \frac{ 498436 }{ 625 } $$ $$ d_1 ^ {\,2} = \frac{ 498436 }{ 625 } - 121 $$ $$ d_1 ^ {\,2} = \frac{ 422811 }{ 625 } $$ $$ d_1 = \sqrt{ \frac{ 422811 }{ 625 } } $$$$ d_1 = \frac{ 3 \sqrt{ 46979}}{ 25 } $$

STEP 2: find angle $ \alpha $

To find angle $ \alpha $ use formula:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ d_2 }{ d_1 } $$

After substituting $ d_2 = 11 $ and $ d_1 = \frac{ 3 \sqrt{ 46979}}{ 25 } $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ 11 }{ \frac{ 3 \sqrt{ 46979}}{ 25 } } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{ 275 \sqrt{ 46979}}{ 140937 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{ 275 \sqrt{ 46979}}{ 140937 } \right) $$ $$ \frac{ \alpha }{ 2 } = 25.0192^o $$$$ \alpha = 25.0192^o \cdot 2 $$$$ \alpha = 50.0383^o $$

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