Find the incircle radius $ r $ of a rhombus if diagonal $d_1 = 14$ and diagonal $d_2 = 7$.

$r = \frac{ 7 }{ 2 }$

STEP 1: find angle $ \alpha $

To find angle $ \alpha $ use formula:

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

After substituting $ d_2 = 7 $ and $ d_1 = 14 $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ 7 }{ 14 } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{ 1 }{ 2 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{ 1 }{ 2 } \right) $$ $$ \frac{ \alpha }{ 2 } = 30^o $$$$ \alpha = 30^o \cdot 2 $$$$ \alpha = 60^o $$

STEP 2: find height $ h $

To find height $ h $ use formula:

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

After substituting $ \alpha = 30 $ and $ d_1 = 14 $ we have:

$$ \sin \left( \frac{ 60^o }{ 2 } \right) = \dfrac{ h }{ 14 } $$ $$ \sin( 30 ) = \dfrac{ h }{ 14 } $$ $$ \frac{ 1 }{ 2 } = \dfrac{ h }{ 14 } $$$$ h = \frac{ 1 }{ 2 } \cdot 14 $$$$ h = 7 $$

STEP 3: find incircle radius $ r $

To find incircle radius $ r $ use formula:

$$ h = 2 \cdot r $$

After substituting $ h = 7 $ we have:

$$ 7 = 2 \cdot r $$ $$ r = \dfrac{ 7 }{ 2 } $$ $$ r = \frac{ 7 }{ 2 } $$

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