Find the height $ h $ of a rhombus if side $a = 120$ and diagonal $d_1 = 212$.

$h = \frac{ 53 \sqrt{ 791}}{ 15 }$

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 = 212 $ and $ a = 120 $ we have:

$$ 212^2 + d_2^2 = 4 \cdot 120^2 $$ $$ 44944 + d_2^2 = 57600 $$ $$ d_2^2 = 57600 - 44944 $$ $$ d_2^2 = 12656 $$ $$ d_2 = \sqrt{ 12656 } $$$$ d_2 = 4 \sqrt{ 791 } $$

STEP 2: find area $ A $

To find area $ A $ use formula:

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

After substituting $ d_1 = 212 $ and $ d_2 = 4 \sqrt{ 791 } $ we have:

$$ A = \dfrac{ 212 \cdot 4 \sqrt{ 791 } }{ 2 }$$$$ A = \dfrac{ 848 \sqrt{ 791 } }{ 2 } $$$$ A = 424 \sqrt{ 791 } $$

STEP 3: find height $ h $

To find height $ h $ use formula:

$$ A = a \cdot h $$

After substituting $ A = 424 \sqrt{ 791 } $ and $ a = 120 $ we have:

$$ 424 \sqrt{ 791 } = 120 \cdot h $$$$ h = \dfrac{ 424 \sqrt{ 791 } }{ 120 } $$$$ h = \frac{ 53 \sqrt{ 791}}{ 15 } $$

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