STEP 1: find diagonal $ d1 $

To find diagonal $ d1 $ use formula:

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

After substituting $d_2 = 5\, \text{cm}$ and $a = 9\, \text{cm}$ we have:

$$ d_1 ^ {\,2} + \left( 5\, \text{cm} \right)^{2} = 4 \cdot \left( 9\, \text{cm} \right)^{2} $$ $$ d_1 ^ {\,2} + 25\, \text{cm}^2 = = 324\, \text{cm}^2 $$ $$ d_1 ^ {\,2} = = 324\, \text{cm}^2 - 25\, \text{cm}^2 $$ $$ d_1 ^ {\,2} = 299\, \text{cm}^2 $$ $$ d_1 = \sqrt{ 299\, \text{cm}^2 } $$$$ d_1 = \sqrt{ 299 }\, \text{cm} $$

STEP 2: find area $ A $

To find area $ A $ use formula:

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

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

$$ A = \dfrac{ \sqrt{ 299 }\, \text{cm} \cdot 5\, \text{cm} }{ 2 }$$$$ A = \dfrac{ 5 \sqrt{ 299 }\, \text{cm}^2 }{ 2 } $$$$ A = \frac{ 5 \sqrt{ 299}}{ 2 }\, \text{cm}^2 $$

STEP 3: find height $ h $

To find height $ h $ use formula:

$$ A = a \cdot h $$

After substituting $A = \dfrac{ 5 \sqrt{ 299}}{ 2 }\, \text{cm}^2$ and $a = 9\, \text{cm}$ we have:

$$ \frac{ 5 \sqrt{ 299}}{ 2 }\, \text{cm}^2 = 9\, \text{cm} \cdot h $$$$ h = \dfrac{ \frac{ 5 \sqrt{ 299}}{ 2 }\, \text{cm}^2 }{ 9\, \text{cm} } $$$$ h = \frac{ 5 \sqrt{ 299}}{ 18 }\, \text{cm} $$

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