2023 · Q42
A line segment AB is bisected at C and semi-circles S1, S2 and S3 are drawn respectively on AB, AC and CB as diameters such that they all lie on same side of AB. A circle S is drawn touching internally S1 and externally S2 and S3. If m is the area of the circle S and n is the area of semi-circle S1, then which one of the following is correct?
Elementary MathematicsCircles
- A9m = 2n
- B9m = 4n
- C3m = 2n
- D7m = 3n
Explanation
From the arbelos result, with AC = CB = 2k the inscribed circle has radius r = 2k/3, so m = πr² = 4πk²/9. The semi-circle S₁ has radius 2k, so n = ½π(2k)² = 2πk². Then 9m = 9·(4πk²/9) = 4πk² = 2·(2πk²) = 2n. The answer is (A) 9m = 2n.