Find the height $ h $ of a rhombus if side $a = 9$ and diagonal $d_2 = 5$.

$h = \frac{ 5 \sqrt{ 299}}{ 18 }$

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 = 5 $ and $ a = 9 $ we have:

$$ d_1 ^ {\,2} + 5^2 = 4 \cdot 9^2 $$ $$ d_1 ^ {\,2} + 25 = 324 $$ $$ d_1 ^ {\,2} = 324 - 25 $$ $$ d_1 ^ {\,2} = 299 $$ $$ d_1 = \sqrt{ 299 } $$

STEP 2: find area $ A $

To find area $ A $ use formula:

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

After substituting $ d_1 = \sqrt{ 299 } $ and $ d_2 = 5 $ we have:

$$ A = \dfrac{ \sqrt{ 299 } \cdot 5 }{ 2 }$$$$ A = \dfrac{ 5 \sqrt{ 299 } }{ 2 } $$$$ A = \frac{ 5 \sqrt{ 299}}{ 2 } $$

STEP 3: find height $ h $

To find height $ h $ use formula:

$$ A = a \cdot h $$

After substituting $ A = \frac{ 5 \sqrt{ 299}}{ 2 } $ and $ a = 9 $ we have:

$$ \frac{ 5 \sqrt{ 299}}{ 2 } = 9 \cdot h $$$$ h = \dfrac{ \frac{ 5 \sqrt{ 299}}{ 2 } }{ 9 } $$$$ h = \frac{ 5 \sqrt{ 299}}{ 18 } $$

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