Find the side $ a $ of a rhombus if diagonal $d_2 = \frac{ 5 }{ 2 }$ and angle $\alpha = 60^o$.

$a = \frac{ 5 \sqrt{ 5}}{ 4 }$

STEP 1: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

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

After substituting $ \alpha = 30^o $ and $ d_2 = \frac{ 5 }{ 2 } $ we have:

$$ \sin \left( \frac{ 60^o }{ 2 } \right) = \dfrac{ d_2 }{ } $$ $$ \sin( 30^o ) = \dfrac{ \frac{ 5 }{ 2 } }{ d_1 } $$ $$ \frac{ 1 }{ 2 } = \dfrac{ \frac{ 5 }{ 2 } }{ d_1 } $$ $$ d_1 = \dfrac{ \frac{ 5 }{ 2 } }{ \frac{ 1 }{ 2 } } $$ $$ d_1 = 5 $$

STEP 2: find side $ a $

To find side $ a $ use formula:

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

After substituting $ d_1 = 5 $ and $ d_2 = \frac{ 5 }{ 2 } $ we have:

$$ 5^2 + \left(\frac{ 5 }{ 2 }\right)^2 = 4 \cdot a^2 $$ $$ 25 + \frac{ 25 }{ 4 } = 4 \cdot a^2 $$ $$ 4 \cdot a^2 = \frac{ 125 }{ 4 } $$ $$ a^2 = \frac{ \frac{ 125 }{ 4 } }{ 4 } $$ $$ a^2 = \frac{ 125 }{ 16 } $$ $$ a = \sqrt{ \frac{ 125 }{ 16 } } $$$$ a = \frac{ 5 \sqrt{ 5}}{ 4 } $$

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