STEP 1: find angle $ \alpha $

To find angle $ \alpha $ use formula:

$$ \alpha + \beta = 180^o $$

After substituting $ \beta = 166^o $ we have:

$$ \alpha + 166^o = 180^o $$ $$ \alpha = 180^o - 166^o $$ $$ \alpha = 14^o $$

STEP 2: find side $ c $

To find side $ c $ use formula:

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

After substituting $\alpha = 14^o$ and $h = 1.5\, \text{cm}$ we have:

$$ \sin( 14^o ) = \dfrac{ 1.5\, \text{cm} }{ c } $$ $$ 0.2419 = \dfrac{ 1.5\, \text{cm} }{ c } $$ $$ c = \dfrac{ 1.5\, \text{cm} }{ 0.2419 } $$ $$ c = 6.2003\, \text{cm} $$

STEP 3: find side $ x $

To find side $ x $ use Pythagorean Theorem:

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

After substituting $h = 1.5\, \text{cm}$ and $c = 6.2003\, \text{cm}$ we have:

$$ \left( 1.5\, \text{cm} \right)^{2} + x^2 = \left( 6.2003\, \text{cm} \right)^{2} $$ $$ x^2 = \left( 6.2003\, \text{cm} \right)^{2} - \left( 1.5\, \text{cm} \right)^{2} $$ $$ x^2 = 38.4443\, \text{cm}^2 - 2.25\, \text{cm}^2 $$ $$ x^2 = 36.1943\, \text{cm}^2 $$ $$ x = \sqrt{ 36.1943\, \text{cm}^2 } $$$$ x = 6.0162\, \text{cm} $$

STEP 4: find long base $ a $

To find long base $ a $ use formula:

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

After substituting $x = 6.0162\, \text{cm}$ and $b = 8\, \text{cm}$ we have:

$$ 6.0162\, \text{cm} = \frac{ a - 8\, \text{cm} } { 2 } $$ $$ 6.0162\, \text{cm} \cdot 2 = a - 8\, \text{cm} $$ $$ a - 8\, \text{cm} = 12.0323\, \text{cm} $$ $$ a = 12.0323\, \text{cm} + 8\, \text{cm} $$ $$ a = 20.0323\, \text{cm} $$

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