# Tripod and a Plane

Geometry Level 5

A tripod is formed by three vectors $$V_1$$, $$V_2$$ and $$V_3$$ that are joined at point $$O$$ as in the figure above. Each vector makes a $$30 ^ \circ$$ angle with the common axis (shown in green), and can be obtained from each other by rotating about the common axis by $$120 ^ \circ$$. Points $$A$$, $$B$$ and $$C$$ are on the lines along $$V_1$$, $$V_2$$ and $$V_3$$ , respectively. A point $$T$$ is on the common axis, and $$10$$ units away from the joining point $$O$$.

If points $$A$$ , $$B$$ , $$C$$ and $$T$$ are coplanar, and if $$x = | OA | = 12$$ and $$y= | OB | = 15$$, what is the value of $$z = | OC |$$ ? (Round your answer to three decimal places)

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