Find the height $ h $ of a rhombus if side $a = 18$ and diagonal $d_2 = 12$.

$h = 8 \sqrt{ 2 }$

STEP 1: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

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

After substituting $ d_2 = 12 $ and $ a = 18 $ we have:

$$ d_1 ^ {\,2} + 12^2 = 4 \cdot 18^2 $$ $$ d_1 ^ {\,2} + 144 = 1296 $$ $$ d_1 ^ {\,2} = 1296 - 144 $$ $$ d_1 ^ {\,2} = 1152 $$ $$ d_1 = \sqrt{ 1152 } $$$$ d_1 = 24 \sqrt{ 2 } $$

STEP 2: find area $ A $

To find area $ A $ use formula:

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

After substituting $ d_1 = 24 \sqrt{ 2 } $ and $ d_2 = 12 $ we have:

$$ A = \dfrac{ 24 \sqrt{ 2 } \cdot 12 }{ 2 }$$$$ A = \dfrac{ 288 \sqrt{ 2 } }{ 2 } $$$$ A = 144 \sqrt{ 2 } $$

STEP 3: find height $ h $

To find height $ h $ use formula:

$$ A = a \cdot h $$

After substituting $ A = 144 \sqrt{ 2 } $ and $ a = 18 $ we have:

$$ 144 \sqrt{ 2 } = 18 \cdot h $$$$ h = \dfrac{ 144 \sqrt{ 2 } }{ 18 } $$$$ h = 8 \sqrt{ 2 } $$

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