# Poly-Poly...Here I come!

Geometry Level 5

Polyhedron $$\text{ABCDEFG}$$ has $$6$$ faces. Face $$\text{ABCD}$$ is a square with $$\text{AB} = 12;$$ face $$\text{ABFG}$$ is a $$\text{trapezoid}$$ with $$\overline{AB}$$ $$\parallel$$ $$\overline{GF},$$ $$BF = AG = 8,$$and $$GF = 6;$$and face $$\text{CDE}$$has $$CE = DE = 14.$$ The other $$3$$ faces are $$ADEG, BCEF,$$ and $$EFG.$$ The distance from $$E$$ to face $$ABCD$$ is $$12$$. Given that $$EG^2 = p - q\sqrt {r},$$ where $$p, q,$$ $$r$$$$\in$$$$\mathbb{I}^+$$, find $$p + q + r.$$

Try my set

($$r$$ is not divisible by the square of any prime) Hint:

Figure looks like this.

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