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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