STEP 1: find side $ a $

To find side $ a $ use formula:

$$ P = 4 \cdot a $$

After substituting $P = 85\, \text{cm}$ we have:

$$ 85\, \text{cm} = 4 \cdot a $$ $$ a = \dfrac{ 85\, \text{cm} }{ 4 } $$ $$ a = \frac{ 85 }{ 4 }\, \text{cm} $$

STEP 2: find height $ h $

To find height $ h $ use formula:

$$ A = a \cdot h $$

After substituting $A = 442\, \text{cm}$ and $a = \dfrac{ 85 }{ 4 }\, \text{cm}$ we have:

$$ 442\, \text{cm} = \frac{ 85 }{ 4 }\, \text{cm} \cdot h $$$$ h = \dfrac{ 442\, \text{cm} }{ \frac{ 85 }{ 4 }\, \text{cm} } $$$$ h = \frac{ 104 }{ 5 } $$

STEP 3: find angle $ \alpha $

To find angle $ \alpha $ use formula:

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

After substituting $h = \dfrac{ 104 }{ 5 }\, \text{cm}^0$ and $a = \dfrac{ 85 }{ 4 }\, \text{cm}$ we have:

$$ \sin \left( \alpha \right) = \dfrac{ \frac{ 104 }{ 5 } }{ \frac{ 85 }{ 4 }\, \text{cm} } $$ $$ \sin \left( \alpha \right) = \frac{ 416 }{ 425 }\, \text{cm}^-1 $$ $$ \alpha = \arcsin\left( \frac{ 416 }{ 425 }\, \text{cm}^-1 \right) $$ $$ \alpha = 78.1877^o $$

STEP 4: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

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

After substituting $\alpha = 78.1877^o$ and $h = \dfrac{ 104 }{ 5 }\, \text{cm}^0$ we have:

$$ \sin \left( \frac{ 78.1877^o }{ 2 } \right) = \dfrac{ h }{ d_1 } $$ $$ \sin( 39.0939 ) = \dfrac{ \frac{ 104 }{ 5 } }{ d_1 } $$ $$ 0.6306 = \dfrac{ \frac{ 104 }{ 5 } }{ d_1 } $$ $$ d_1 = \dfrac{ \frac{ 104 }{ 5 } }{ 0.6306 } $$ $$ d_1 = 32.9848 $$

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