Two tangents (AB and AC) are drawn to the circle centered at point O to point A, the angle between

| September 7, 2021 | education

Two tangents (AB and AC) are drawn to the circle centered at point O to point A, the angle between which is 120 (degrees). Find the lengths of the tangent line segments (AB and AC) if OA is 24cm.

From point O we draw the radii OB and OС to the points of tangency B and C.

The radii OB and OS are perpendicular to AB and AC, then the triangles AOB and AOC are rectangular, in which the hypotenuse of OA is common, and the legs OB = OС = R.

Then the triangles AOB and AOC are equal in leg and hypotenuse, which means that the angle OAC = OAB = 120/2 = 60, then the angle AOB = AOC = 90 – 60 = 30.

The AC leg lies against an angle of 30, then AC = OA / 2 = 24/2 = 12 cm.

Since the tangents AB and AC are drawn from the same point, then AB = AC = 12 cm.

Answer: The lengths of the tangents are 12 cm.



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