If r is the radius of S and R is the radius of S_2, then which one of the following is <strong>CORRECT</strong>?
Consider the following for the next two (02) items that follow : A line segment AB is bisected at C and semi-circles S_1, S_2 and S_3 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 S_1 and externally S_2 and S_3.
- A. R=3r
- B. R=2r
- C. 3R=4r
- D. 2R=3r ✓
Correct Answer: D. 2R=3r
Explanation
Radius of S_1 is 2R. Radius of S_2, S_3 is R. Let S have radius r. The center of S forms a right triangle with the center of S_2 and the center of S_1. The hypotenuse is R+r, base is R, and height is 2R-r. By Pythagoras theorem, (2R-r)^2 + R^2 = (R+r)^2, which simplifies to 4R^2 = 6Rr \implies 2R = 3r.
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