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

Refer to the attached diagram for help.

All radii of a circle are congruent.
Tangent segs LP and LQ are congruent. (Tangent segs drawn to a circle from the same point are congruent)

Because of this we can establish OPLQ as a kite.

OPLQ is a kite implies that OL bis. angle PLQ.
An angle bis. divides an angle into 2 congruent, parts, so angle OLP must be 30 degrees.

Since tangents form right angles with the radius, angle OPL is right.

Now we have a right triangle OPL with a 30-degree angle. The other angle must be 60 degrees because of the no-choice theorem. (180-(30+90)) = 60
We know that OP is 6 because it is a radius of circle O. How do we use that to find OL? Well, we could use trig, but you might recognize this as a special triangle!

The side opposite the 30 degree angle = x
The side opposite the right angle = 2x

So OL must be 12!