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$$\text {Trapezoid PLAY}$$

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A playground company, Triangles-R-Us, has drafted a framework design for a multi-purpose play area for children. The triangles on the right and left of the design are used as the framework for slides or tents with the top of the rectangular frame used as a lateral ladder or as a bar for hanging swings.

Calculate the exact footage of metal framing needed to construct the presented framework.

Note: $\overline {PY}$ denotes the ground. No metal framing is needed. All remaining segments are considered metal framing.


$9+8\sqrt {2}+\cfrac {16\sqrt {3}}{3}$


$25+8\sqrt {2}+\cfrac {16\sqrt {3}}{3}$


$33\sqrt {2}+\cfrac {16\sqrt {3}}{3}$


$9+8\sqrt {2}+\cfrac {8\sqrt {3}}{3}$


$42+8\sqrt {2}+\frac {24\sqrt {3}}{3}$

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