Find the diagonal $ d_1 $ of a rhombus if diagonal $d_2 = \frac{ 6771 }{ 500 }$ and incircle radius $r = \frac{ 6371 }{ 500 }$.

Problem has no solutions.

STEP 1: find height $ h $

To find height $ h $ use formula:

$$ h = 2 \cdot r $$

After substituting $ r = \frac{ 6371 }{ 500 } $ we have:

$$ h = 2 \cdot \frac{ 6371 }{ 500 } $$ $$ h = \frac{ 6371 }{ 250 } $$

STEP 2: find angle $ \beta $

To find angle $ \beta $ use formula:

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

After substituting $ h = \frac{ 6371 }{ 250 } $ and $ d_2 = \frac{ 6771 }{ 500 } $ we have:

$$ \sin \left( \frac{ \beta }{ 2 } \right) = \dfrac{ \frac{ 6371 }{ 250 } }{ \frac{ 6771 }{ 500 } } $$ $$ \sin \left( \frac{ \beta }{ 2 } \right) = \frac{ 12742 }{ 6771 } $$ $$ \frac{ \beta }{ 2 } = \arcsin\left( \frac{ 12742 }{ 6771 } \right) $$

$ \arcsin(1.882) $ is not defined $ \Longrightarrow $ The problem has no solution.

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