If a straight line intersects two concentric circles with center at the points and in that order, what is the geometric relationship between the segment lengths and ?
Hint 1: Draw a line from the common center perpendicular to the secant line.
Hint 2: Use the theorem that a perpendicular from the center of a circle to any chord bisects that chord.
Hint 3: Relate the bisected segments of the outer circle () and the inner circle () by subtraction.
Step 1 (Draw a perpendicular): Let be the common center of the concentric circles. Draw a perpendicular line segment from to the intersecting line, meeting it at point . Thus, .
Step 2 (Apply chord bisector theorem): A perpendicular from the center of a circle to a chord bisects the chord:
For the larger circle, is a chord, so bisects . Therefore:
For the smaller circle, is a chord, so bisects . Therefore:
Step 3 (Subtract segments): Subtract Equation 2 from Equation 1:
Step 4 (Conclusion): The relationship is exactly AB = CD.
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