AB and AC are segments of tangents drawn to a circle of radius 9 cm. Find the lengths of segments AC and AO if AB = 12cm.

To solve the problem, we will use the properties of tangents to the circle.
By the condition of the problem, the segments of the tangents are drawn from one point – point A, therefore, AB = AC = 12 cm.
The second property is that at the point of tangency, the tangent line is perpendicular to the radius. We get AB ⊥OB and AC ⊥ OS.
Two right-angled triangles ABO and ASO are equal (legs are equal, hypotenuse is common).
By the Pythagorean theorem, we find the hypotenuse of AO.
AO = √ (AB² + OB²) = √ (144 + 81) = √225 = 15 (cm).
Answer: the length of the AU is 12 cm, the length of the AO is 15 cm.