Question

From exterior point L of circle O, tangent segments LP and LQ are drawn such that the measure of angle PLQ is 60 degrees. If a radius of circle O measures 6, what is the distance from the center of the circle to point L?

Answer 1

Wow !  There's so much extra mush here that the likelihood of being
distracted and led astray is almost unavoidable.

The circle ' O '  is roughly 98.17% (π/3.2) useless to us.  The only reason
we need it at all is in order to recall that the tangent to a circle is
perpendicular to the radius drawn to the tangent point.  And now
we can discard Circle - ' O ' .
Just keep the point at its center, and call it point - O .

-- The segments LP, LQ, and LO, along with the radii OP and OQ, form
two right triangles, reposing romantically hypotenuse-to-hypotenuse. 
The length of segment LO ... their common hypotenuse ... is the answer
to the question.

-- Angle PLQ is 60 degrees.  The common hypotenuse is its bisector.
So the acute angle of each triangle at point ' L ' is 30 degrees, and the
acute angle of each triangle at point ' O ' is 60 degrees.

-- The leg of each triangle opposite the 30-degree angle is a radius
of the discarded circle, and measures 6 .

-- In every 30-60 right triangle, the length of the side opposite the hypotenuse
is  one-half the length of the hypotenuse.

-- So the length of the hypotenuse (segment LO) is  12 .