Find the angle $ \beta $ of a rhombus if area $A = 143$ and diagonal $d_1 = 26$.

$\beta = 39.942^o$

STEP 1: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

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

After substituting $ A = 143 $ and $ d_1 = 26 $ we have:

$$ 143 = \dfrac{ 26 \cdot d_2 }{ 2 } $$$$ 143 \cdot 2 = 26 \cdot d_2 $$$$ 286 = 26 \cdot d_2 $$$$ d_2 = \dfrac{ 286 }{ 26 } $$$$ d_2 = 11 $$

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 = 11 $ and $ d_1 = 26 $ we have:

$$ \sin \left( \frac{ \alpha }{ 2 } \right) = \dfrac{ 11 }{ 26 } $$ $$ \sin \left( \frac{ \alpha }{ 2 } \right) = \frac{ 11 }{ 26 } $$ $$ \frac{ \alpha }{ 2 } = \arcsin\left( \frac{ 11 }{ 26 } \right) $$ $$ \frac{ \alpha }{ 2 } = 25.029^o $$$$ \alpha = 25.029^o \cdot 2 $$$$ \alpha = 50.058^o $$

STEP 3: find angle $ \beta $

To find angle $ \beta $ use formula:

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

After substituting $ \alpha = 50.058^o $ we have:

$$ 50.058^o + \beta = 90^o $$ $$ \beta = 90^o - 50.058^o $$ $$ \beta = 39.942^o $$

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