Find the incircle radius $ r $ of a rhombus if diagonal $d_1 = 8$ and diagonal $d_2 = 6$.

$r = 3$

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 = 6 $ and $ d_1 = 8 $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ 6 }{ 8 } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{ 3 }{ 4 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{ 3 }{ 4 } \right) $$ $$ \frac{ \alpha }{ 2 } = 48.5904^o $$$$ \alpha = 48.5904^o \cdot 2 $$$$ \alpha = 97.1808^o $$

STEP 2: find height $ h $

To find height $ h $ use formula:

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

After substituting $ \alpha = 48.5904 $ and $ d_1 = 8 $ we have:

$$ \sin \left( \frac{ 97.1808^o }{ 2 } \right) = \dfrac{ h }{ 8 } $$ $$ \sin( 48.5904 ) = \dfrac{ h }{ 8 } $$ $$ 0.75 = \dfrac{ h }{ 8 } $$$$ h = 0.75 \cdot 8 $$$$ h = 6 $$

STEP 3: find incircle radius $ r $

To find incircle radius $ r $ use formula:

$$ h = 2 \cdot r $$

After substituting $ h = 6 $ we have:

$$ 6 = 2 \cdot r $$ $$ r = \dfrac{ 6 }{ 2 } $$ $$ r = 3 $$

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