Find the lateral edge $ e $ of a pyramid if height $h = 16$ and slant height $s = 20$.

$e = 4 \sqrt{ 34 }$

STEP 1: find side $ a $

To find side $ a $ use Pythagorean Theorem:

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

After substituting $ h = 16 $ and $ s = 20 $ we have:

$$ 16^2 + \frac{ a^2 }{ 4 } = 20^2 $$ $$ \frac{ a^2 }{ 4 } = 20^2 - 16^2 $$ $$ \frac{ a^2 }{ 4 } = 400 - 256 $$ $$ a^2 = 144 \cdot 4 $$ $$ a^2 = 576 $$ $$ a = \sqrt{ 576 } $$$$ a = 24 $$

STEP 2: find lateral edge $ e $

To find lateral edge $ e $ use Pythagorean Theorem:

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

After substituting $ s = 20 $ and $ a = 24 $ we have:

$$ 20^2 + \frac{ 24^2 }{ 4 }= e^2 $$ $$ 400 + \frac{ 576 }{ 4 }= e^2 $$ $$ 400 + 144 = e^2 $$ $$ e^2 = 544 $$ $$ e = \sqrt{ 544 } $$$$ e = 4 \sqrt{ 34 } $$

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