Find the incircle radius $ r $ of a rhombus if diagonal $d_2 = 13542$ and angle $\beta = 170^o$.

$r = 6745.2343$

STEP 1: find height $ h $

To find height $ h $ use formula:

$$ \sin \left( \frac{ \beta }{ 2 } \right) = \dfrac{ h }{ d_2 } $$

After substituting $ \beta = 85^o $ and $ d_2 = 13542 $ we have:

$$ \sin \left( \frac{ 170^o }{ 2 } \right) = \dfrac{ h }{ 13542 } $$ $$ \sin( 85^o ) = \dfrac{ h }{ 13542 } $$ $$ 0.9962 = \dfrac{ h }{ 13542 } $$$$ h = 0.9962 \cdot 13542 $$$$ h = 13490.4686 $$

STEP 2: find incircle radius $ r $

To find incircle radius $ r $ use formula:

$$ h = 2 \cdot r $$

After substituting $ h = 13490.4686 $ we have:

$$ 13490.4686 = 2 \cdot r $$ $$ r = \dfrac{ 13490.4686 }{ 2 } $$ $$ r \approx 6745.2343 $$

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