STEP 1: find angle $ \alpha $

To find angle $ \alpha $ use formula:

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

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

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

STEP 2: find height $ h $

To find height $ h $ use formula:

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

After substituting $\alpha = 60^o$ and $d_1 = 3\, \text{cm}$ we have:

$$ \sin \left( \frac{ 60^o }{ 2 } \right) = \dfrac{ h }{ d_1 } $$ $$ \sin( 30^o ) = \dfrac{ h }{ 3 } $$ $$ \frac{ 1 }{ 2 } = \dfrac{ h }{ 3 } $$$$ h = \frac{ 1 }{ 2 } \cdot 3 $$$$ h = \frac{ 3 }{ 2 } $$

STEP 3: find side $ a $

To find side $ a $ use formula:

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

After substituting $\alpha = 60^o$ and $h = \dfrac{ 3 }{ 2 }\, \text{cm}$ we have:

$$ \sin( 60^o ) = \dfrac{ \frac{ 3 }{ 2 }\, \text{cm} }{ a } $$ $$ \frac{\sqrt{ 3 }}{ 2 } = \dfrac{ \frac{ 3 }{ 2 }\, \text{cm} }{ a } $$ $$ a = \dfrac{ \frac{ 3 }{ 2 }\, \text{cm} }{ \frac{\sqrt{ 3 }}{ 2 } } $$ $$ a = \sqrt{ 3 }\, \text{cm} $$

STEP 4: find perimeter $ P $

To find perimeter $ P $ use formula:

$$ P = 4 \cdot a $$

After substituting $a = \sqrt{ 3 }\, \text{cm}$ we have:

$$ P = 4 \cdot \sqrt{ 3 }\, \text{cm} $$ $$ P = 4 \sqrt{ 3 }\, \text{cm} $$

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