STEP 1: find short base $ b $
To find short base $ b $ use formula:
$$ A = \frac{ (a + b) \cdot h}{ 2 }$$After substituting $ A = 52 $ , $ a = 10 $ and $ h = 6 $ we have:
$$ 52 = \frac{ (10 + b) \cdot 6}{ 2 }$$$$ 52 \cdot 2 = (10 + b) \cdot 6 $$$$ 104 = (10 + b) \cdot 6 $$$$ 10 + b = \frac{ 104 }{ 6 } $$$$ 10 + b = \frac{ 52 }{ 3 } $$$$ b = \frac{ 52 }{ 3 } - 10 $$$$ a = \frac{ 22 }{ 3 } $$STEP 2: find side $ y $
To find side $ y $ use formula:
$$ y = \frac{ a + b } { 2 } $$After substituting $ a = 10 $ and $ b = \frac{ 22 }{ 3 } $ we have:
$$ y = \frac{ 10 + \frac{ 22 }{ 3 } } { 2 } $$ $$ y = \frac{ \frac{ 52 }{ 3 } } { 2 } $$ $$ y = \frac{ 26 }{ 3 } $$STEP 3: find diagonal $ d $
To find diagonal $ d $ use Pythagorean Theorem:
$$ h^2 + y^2 = d^2 $$After substituting $ h = 6 $ and $ y = \frac{ 26 }{ 3 } $ we have:
$$ 6^2 + \left(\frac{ 26 }{ 3 }\right)^2 = d^2 $$ $$ 36 + \frac{ 676 }{ 9 } = d^2 $$ $$ d^2 = \frac{ 1000 }{ 9 } $$ $$ d = \sqrt{ \frac{ 1000 }{ 9 } } $$$$ d = \frac{ 10 \sqrt{ 10}}{ 3 } $$