From point A, two tangents are drawn to a circle with a center at point O and a radius of 7 cm. Find the angle between them if AO = 14 cm.
Let two tangents be drawn from point A to the circle, they touch the circle at points B and C, by the property of the tangent drawn from the point they are equal: AB = AC. A straight line connecting point A with the center of the circle O, AO = 14 cm, <CAB = <CAO + <BAO
Consider a triangle AOB, <ABO = 90 °, the tangent is perpendicular to the radius drawn to the point of tangency, which means AOB is rectangular with legs AB and BO and hypotenuse AO, AO = 14 cm, BO = R = 7 cm, according to the Pythagorean theorem, we find the second leg :
AB = √ (AO²-BO²) = √ (14²-7²) = √147.
Find the angle <BAO, through the sine:
sin BAO = BO / AO = 7/14 = 1/2.
According to the table, we determine the angle <BAO = 30 °.
Let’s compare the triangles AOC and AOB, AC = AB, AO = AO, BO = BC, so they are equal, and their corresponding angles are equal:
<СAO <BAO = 30 °.
<CAB = <CAO + <BAO = 30 + 30 = 60 °.
Answer: the angle between tangents is 60 °.
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