STEP 1: find side $ a $

To find side $ a $ use formula:

$$ d = \sqrt{ 2 } \cdot a $$

After substituting $d = 10\, \text{cm}$ we have:

$$ 10\, \text{cm} = \sqrt{ 2 } \cdot a $$ $$ a = \dfrac{ 10\, \text{cm} }{ \sqrt{ 2 } } $$ $$ a = 5 \sqrt{ 2 }\, \text{cm} $$

STEP 2: find slant height $ s $

To find slant height $ s $ use Pythagorean Theorem:

$$ h^2 + \frac{ a^2 }{ 4 } = s^2 $$

After substituting $h = 12\, \text{cm}$ and $a = 5 \sqrt{ 2 }\, \text{cm}$ we have:

$$ \left( 12\, \text{cm} \right)^{2} + \frac{ \left( 5 \sqrt{ 2 }\, \text{cm} \right)^{2} }{ 4 }= s^2 $$ $$ 144\, \text{cm}^2 + \frac{ 50\, \text{cm}^2 }{ 4 }= s^2 $$ $$ 144\, \text{cm}^2 + \frac{ 25 }{ 2 }\, \text{cm}^2 = s^2 $$ $$ s^2 = \frac{ 313 }{ 2 }\, \text{cm}^2 $$ $$ s = \sqrt{ \frac{ 313 }{ 2 }\, \text{cm}^2 } $$$$ s = \frac{\sqrt{ 626 }}{ 2 }\, \text{cm} $$

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