Find the perimeter $ P $ of a rhombus if diagonal $d_2 = 25$ and angle $\beta = 60^o$.

$P = 100$

STEP 1: find angle $ \alpha $

To find angle $ \alpha $ use formula:

$$ \alpha + \beta = 90^o $$

After substituting $ \beta = 60^o $ we have:

$$ \alpha + 60^o = 90^o $$ $$ \alpha = 90^o - 60^o $$ $$ \alpha = 30^o $$

STEP 2: find height $ h $

To find height $ h $ use formula:

$$ \sin \left( \frac{ \beta }{ 2 } \right) = \dfrac{ h }{ d_2 } $$

After substituting $ \beta = 30^o $ and $ d_2 = 25 $ we have:

$$ \sin \left( \frac{ 60^o }{ 2 } \right) = \dfrac{ h }{ 25 } $$ $$ \sin( 30^o ) = \dfrac{ h }{ 25 } $$ $$ \frac{ 1 }{ 2 } = \dfrac{ h }{ 25 } $$$$ h = \frac{ 1 }{ 2 } \cdot 25 $$$$ h = \frac{ 25 }{ 2 } $$

STEP 3: find side $ a $

To find side $ a $ use formula:

$$ \sin \left( \alpha \right) = \dfrac{ h }{ a } $$

After substituting $ \alpha = 30^o $ and $ h = \frac{ 25 }{ 2 } $ we have:

$$ \sin( 30^o ) = \dfrac{ \frac{ 25 }{ 2 } }{ a } $$ $$ \frac{ 1 }{ 2 } = \dfrac{ \frac{ 25 }{ 2 } }{ a } $$ $$ a = \dfrac{ \frac{ 25 }{ 2 } }{ \frac{ 1 }{ 2 } } $$ $$ a = 25 $$

STEP 4: find perimeter $ P $

To find perimeter $ P $ use formula:

$$ P = 4 \cdot a $$

After substituting $ a = 25 $ we have:

$$ P = 4 \cdot 25 $$ $$ P = 100 $$

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