Find the angle $ \beta $ of a rhombus if perimeter $P = 80$ and diagonal $d_2 = 24$.

Problem has no solutions.

STEP 1: find side $ a $

To find side $ a $ use formula:

$$ P = 4 \cdot a $$

After substituting $ P = 80 $ we have:

$$ 80 = 4 \cdot a $$ $$ a = \dfrac{ 80 }{ 4 } $$ $$ a = 20 $$

STEP 2: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

$$ d_1^2 + d_2^2 = 4 \cdot a^2 $$

After substituting $ d_2 = 24 $ and $ a = 20 $ we have:

$$ d_1 ^ {\,2} + 24^2 = 4 \cdot 20^2 $$ $$ d_1 ^ {\,2} + 576 = 1600 $$ $$ d_1 ^ {\,2} = 1600 - 576 $$ $$ d_1 ^ {\,2} = 1024 $$ $$ d_1 = \sqrt{ 1024 } $$$$ d_1 = 32 $$

STEP 3: find angle $ \alpha $

To find angle $ \alpha $ use formula:

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

After substituting $ d_2 = 24 $ and $ d_1 = 32 $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ 24 }{ 32 } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{ 3 }{ 4 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{ 3 }{ 4 } \right) $$ $$ \frac{ \alpha }{ 2 } = 48.5904^o $$$$ \alpha = 48.5904^o \cdot 2 $$$$ \alpha = 97.1808^o $$

STEP 4: find angle $ \beta $

To find angle $ \beta $ use formula:

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

After substituting $ \alpha = 97.1808^o $ we have:

$$ 97.1808^o + \beta = 90^o $$ $$ \beta = 90^o - 97.1808^o $$ $$ \beta = -7.1808^o $$

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

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