STEP 1: find height $ h $

To find height $ h $ use formula:

$$ \sin \left( \alpha \right) = \dfrac{ h }{ c } $$

After substituting $\alpha = 45^o$ and $c = 9\, \text{cm}$ we have:

$$ \sin( 45^o ) = \dfrac{ h }{ 9 } $$ $$ \frac{\sqrt{ 2 }}{ 2 } = \dfrac{ h }{ 9 } $$$$ h = \frac{\sqrt{ 2 }}{ 2 } \cdot 9 $$$$ h = \frac{ 9 \sqrt{ 2}}{ 2 } $$

STEP 2: find side $ x $

To find side $ x $ use Pythagorean Theorem:

$$ h^2 + x^2 = c^2 $$

After substituting $h = \dfrac{ 9 \sqrt{ 2}}{ 2 }\, \text{cm}$ and $c = 9\, \text{cm}$ we have:

$$ \left( \frac{ 9 \sqrt{ 2}}{ 2 }\, \text{cm} \right)^{2} + x^2 = \left( 9\, \text{cm} \right)^{2} $$ $$ x^2 = \left( 9\, \text{cm} \right)^{2} - \left( \frac{ 9 \sqrt{ 2}}{ 2 }\, \text{cm} \right)^{2} $$ $$ x^2 = 81\, \text{cm}^2 - \frac{ 81 }{ 2 }\, \text{cm}^2 $$ $$ x^2 = \frac{ 81 }{ 2 }\, \text{cm}^2 $$ $$ x = \sqrt{ \frac{ 81 }{ 2 }\, \text{cm}^2 } $$$$ x = \frac{ 9 \sqrt{ 2}}{ 2 }\, \text{cm} $$

STEP 3: find short base $ b $

To find short base $ b $ use formula:

$$ x = \frac{ a - b } { 2 } $$

After substituting $x = \dfrac{ 9 \sqrt{ 2}}{ 2 }\, \text{cm}$ and $a = 31\, \text{cm}$ we have:

$$ \frac{ 9 \sqrt{ 2}}{ 2 }\, \text{cm} = \frac{ 31\, \text{cm} - b } { 2 } $$ $$ \frac{ 9 \sqrt{ 2}}{ 2 }\, \text{cm} \cdot 2 = 31\, \text{cm} - b $$ $$ 31\, \text{cm} - b = 9 \sqrt{ 2 }\, \text{cm} $$ $$ b = 31\, \text{cm} - 9 \sqrt{ 2 }\, \text{cm} $$ $$ b = 18.2721\, \text{cm} $$

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