2021 Fall AMC 12A Problems/Problem 11
Problem
Consider two concentric circles of radius and The larger circle has a chord, half of which lies inside the smaller circle. What is the length of the chord in the larger circle?
Solution (Power of a Point)
Draw the diameter perpendicular to the chord. Call the intersection between that diameter and the chord In the circle of radius , let the shorter piece of the diameter cut by the chord would be of length , making the longer piece In that same circle, let the be the length of the portion of the chord in the smaller circle that is cut by the diameter we drew. Thus, in the circle of radius , the shorter piece of the diameter cut by the chord would be of length , making the longer piece and length of the piece of the chord cut by the diameter would be (as given in the problem statement). By Power of a Point, we can construct the system of equations Expanding both equations, we get and in which the and terms magically cancel when we subtract the first equation from the second equation. Thus, now we have
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