Lines pass through point A and touch the circle at points B and C. It is known that OB = 4 cm
Lines pass through point A and touch the circle at points B and C. It is known that OB = 4 cm and AB = 4√3. Subtract the angle BAC.
1. From the properties of tangents it is known that the tangents drawn from one point to the circle are equal and form equal angles with the straight line connecting this point and the center of the circle.
AB = AC; angle OAB = angle OAC = angle BAC / 2.
2. Consider a triangle OBA: the angle OBA = 90 degrees, since the radius drawn to the point of tangency is perpendicular to the tangent, OB = 4 cm and AB = 4√3 are the legs.
The tangent of an acute angle in a right-angled triangle is the ratio of the opposite leg to the adjacent one, then:
tgOAB = OB / AB;
tgOAB = 4 / 4√3 = 1 / √3 = √3 / 3.
√3 / 3 is the tangent of an angle equal to 30 degrees, then:
angle ОАВ = angle ОАС = 30 degrees.
3. Angle BAC = 2 * angle OAB;
angle BAC = 2 * 30 = 60 (degrees).
Answer: angle BAC = 60 degrees.