Find the side $ a $ of a rhombus if diagonal $d_2 = 13$ and height $h = 12$.

Problem has no solutions.

STEP 1: find angle $ \beta $

To find angle $ \beta $ use formula:

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

After substituting $ h = 12 $ and $ d_2 = 13 $ we have:

$$ \sin \left( \frac{ \beta }{ 2 } \right) = \dfrac{ 12 }{ 13 } $$ $$ \sin \left( \frac{ \beta }{ 2 } \right) = \frac{ 12 }{ 13 } $$ $$ \frac{ \beta }{ 2 } = \arcsin\left( \frac{ 12 }{ 13 } \right) $$ $$ \frac{ \beta }{ 2 } = 67.3801^o $$$$ \beta = 67.3801^o \cdot 2 $$$$ \beta = 134.7603^o $$

STEP 2: find angle $ \alpha $

To find angle $ \alpha $ use formula:

$$ \alpha + \beta = 90^o $$

After substituting $ \beta = 134.7603^o $ we have:

$$ \alpha + 134.7603^o = 90^o $$ $$ \alpha = 90^o - 134.7603^o $$ $$ \alpha = -44.7603^o $$

The result has to be greater than zero. $ \Longrightarrow $ The problem has no solution.

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