Two concentric circles have radii of 4 cm and 8 cm. A segment tangent to the smaller circle is a chord of the larger circle. What is the length of the segment? (answer in simplest radical form)

Answer 1

This problem can be solved by realizing that the radius of the small circle is perpendicular to the tangent to it at that point, which is a chord of the larger circle. Hence, a right triangle is formed with the radius of the large circle being the hypotenuse, the radius of the small circle being one leg, and HALF the tangent/chord being the other leg. Since the radii of the two circles are given as 4 and 8 centimeters, the half-segment that is the other leg must have a length of 4√3 centimeters, and thus the whole segment has a length of 8√3 centimeters.