Find the height $ h $ of a rhombus if area $A = 500$ and diagonal $d_1 = 25$.

Problem has no solutions.

STEP 1: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

$$ A = \dfrac{ d_1 \cdot d_2 }{ 2 } $$

After substituting $ A = 500 $ and $ d_1 = 25 $ we have:

$$ 500 = \dfrac{ 25 \cdot d_2 }{ 2 } $$$$ 500 \cdot 2 = 25 \cdot d_2 $$$$ 1000 = 25 \cdot d_2 $$$$ d_2 = \dfrac{ 1000 }{ 25 } $$$$ d_2 = 40 $$

STEP 2: find angle $ \alpha $

To find angle $ \alpha $ use formula:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ d_2 }{ d_1 } $$

After substituting $ d_2 = 40 $ and $ d_1 = 25 $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ 40 }{ 25 } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{ 8 }{ 5 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{ 8 }{ 5 } \right) $$

$ \arcsin(1.6) $ is not defined $ \Longrightarrow $ The problem has no solution.

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