Find the height $ h $ of a rhombus if side $a = \frac{ 504 }{ 25 }$ and diagonal $d_1 = 35$.

$h = \frac{ 35 \sqrt{ 5111}}{ 144 }$

STEP 1: find diagonal $ d2 $

To find diagonal $ d2 $ use formula:

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

After substituting $ d_1 = 35 $ and $ a = \frac{ 504 }{ 25 } $ we have:

$$ 35^2 + d_2^2 = 4 \cdot \left(\frac{ 504 }{ 25 }\right)^2 $$ $$ 1225 + d_2^2 = \frac{ 1016064 }{ 625 } $$ $$ d_2^2 = \frac{ 1016064 }{ 625 } - 1225 $$ $$ d_2^2 = \frac{ 250439 }{ 625 } $$ $$ d_2 = \sqrt{ \frac{ 250439 }{ 625 } } $$$$ d_2 = \frac{ 7 \sqrt{ 5111}}{ 25 } $$

STEP 2: find area $ A $

To find area $ A $ use formula:

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

After substituting $ d_1 = 35 $ and $ d_2 = \frac{ 7 \sqrt{ 5111}}{ 25 } $ we have:

$$ A = \dfrac{ 35 \cdot \frac{ 7 \sqrt{ 5111}}{ 25 } }{ 2 }$$$$ A = \dfrac{ \frac{ 49 \sqrt{ 5111}}{ 5 } }{ 2 } $$$$ A = \frac{ 49 \sqrt{ 5111}}{ 10 } $$

STEP 3: find height $ h $

To find height $ h $ use formula:

$$ A = a \cdot h $$

After substituting $ A = \frac{ 49 \sqrt{ 5111}}{ 10 } $ and $ a = \frac{ 504 }{ 25 } $ we have:

$$ \frac{ 49 \sqrt{ 5111}}{ 10 } = \frac{ 504 }{ 25 } \cdot h $$$$ h = \dfrac{ \frac{ 49 \sqrt{ 5111}}{ 10 } }{ \frac{ 504 }{ 25 } } $$$$ h = \frac{ 35 \sqrt{ 5111}}{ 144 } $$

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