Two tangents to a circle with center at point O are drawn from point A. Find the radius of the circle if the angle between the tangents is 60 °, and the distance from point A to point O is 6.
We draw a segment AO, this segment is 6 (by the condition of the problem).
We denote one of the points of tangency of the circle and the tangent as P.
Draw a segment OP.
OR is a perpendicular to the tangent of the AR (by the property of the tangent).
Consider the triangle AOR. This triangle is rectangular, because OR is perpendicular to AR.
AO is the bisector of an angle formed by tangents (a property of tangent lines).
Accordingly, the angle of the radioactive waste is equal to half this angle, i.e. 30 °.
sin∠PAO = sin30 ° = 1/2 (tabular value)
sin∠PAO = OP / AO (by definition of the sine).
1/2 = OR / AO
OP = AO / 2 = 6/2 = 3
This is the radius of the circle.
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