Find the diagonal $ d_1 $ of a rhombus if perimeter $P = 32$ and angle $\alpha = 45^o$.

$d_1 = 14.7821$

STEP 1: find side $ a $

To find side $ a $ use formula:

$$ P = 4 \cdot a $$

After substituting $ P = 32 $ we have:

$$ 32 = 4 \cdot a $$ $$ a = \dfrac{ 32 }{ 4 } $$ $$ a = 8 $$

STEP 2: find height $ h $

To find height $ h $ use formula:

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

After substituting $ \alpha = 45^o $ and $ a = 8 $ we have:

$$ \sin( 45^o ) = \dfrac{ h }{ 8 } $$ $$ \frac{\sqrt{ 2 }}{ 2 } = \dfrac{ h }{ 8 } $$$$ h = \frac{\sqrt{ 2 }}{ 2 } \cdot 8 $$$$ h = 4 \sqrt{ 2 } $$

STEP 3: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

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

After substituting $ \alpha = \frac{ 45 }{ 2 }^o $ and $ h = 4 \sqrt{ 2 } $ we have:

$$ \sin \left( \frac{ 45^o }{ 2 } \right) = \dfrac{ h }{ } $$ $$ \sin( \frac{ 45 }{ 2 }^o ) = \dfrac{ 4 \sqrt{ 2 } }{ d_1 } $$ $$ 0.3827 = \dfrac{ 4 \sqrt{ 2 } }{ d_1 } $$ $$ d_1 = \dfrac{ 4 \sqrt{ 2 } }{ 0.3827 } $$ $$ d_1 = 14.7821 $$

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