Find the angle $ \beta $ of a rhombus if diagonal $d_1 = 36$ and incircle radius $r = 18$.
Answer
Problem has no solutions.
Explanation
STEP 1: find height $ h $
To find height $ h $ use formula:
$$ h = 2 \cdot r $$After substituting $ r = 18 $ we have:
$$ h = 2 \cdot 18 $$ $$ h = 36 $$STEP 2: find angle $ \alpha $
To find angle $ \alpha $ use formula:
$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ h }{ d_1 } $$After substituting $ h = 36 $ and $ d_1 = 36 $ we have:
$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ 36 }{ 36 } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = 1 $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( 1 \right) $$ $$ \frac{ \alpha }{ 2 } = 90^o $$Angle $ $ \alpha $ $ has to be lesser than 90 degrees $ \Longrightarrow $ The problem has no solution.
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