Find the height $ h $ of a rhombus if area $A = 480$ and diagonal $d_1 = 48$.

$h = 20$

STEP 1: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

$$ A = \dfrac{ d_1 \cdot d_2 }{ 2 } $$

After substituting $ A = 480 $ and $ d_1 = 48 $ we have:

$$ 480 = \dfrac{ 48 \cdot d_2 }{ 2 } $$$$ 480 \cdot 2 = 48 \cdot d_2 $$$$ 960 = 48 \cdot d_2 $$$$ d_2 = \dfrac{ 960 }{ 48 } $$$$ d_2 = 20 $$

STEP 2: find angle $ \alpha $

To find angle $ \alpha $ use formula:

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

After substituting $ d_2 = 20 $ and $ d_1 = 48 $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ 20 }{ 48 } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{ 5 }{ 12 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{ 5 }{ 12 } \right) $$ $$ \frac{ \alpha }{ 2 } = 24.6243^o $$$$ \alpha = 24.6243^o \cdot 2 $$$$ \alpha = 49.2486^o $$

STEP 3: find height $ h $

To find height $ h $ use formula:

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

After substituting $ \alpha = 24.6243 $ and $ d_1 = 48 $ we have:

$$ \sin \left( \frac{ 49.2486^o }{ 2 } \right) = \dfrac{ h }{ 48 } $$ $$ \sin( 24.6243 ) = \dfrac{ h }{ 48 } $$ $$ 0.4167 = \dfrac{ h }{ 48 } $$$$ h = 0.4167 \cdot 48 $$$$ h = 20 $$

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