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    Question

    Three circles each of radius 5 cm touch one another. The

    area (in cm²) subtended between them.
    A 25 (√3+ π/2 Correct Answer Incorrect Answer
    B 50 (√3- π/2) Correct Answer Incorrect Answer
    C 25 (2√3- π/2) Correct Answer Incorrect Answer
    D 25 (√3- π/2) Correct Answer Incorrect Answer

    Solution

    Each side of the triangle formed by the centers of the circles is equal to the sum of the radii of two touching circles, which is 2 x 5 = 10 cm. The formula for the area of an equilateral triangle is- Area = (√3/4) a2 Where s is the side length of the triangle. Here, a = 10 cm. A = √3/ 4 × 10² = 25√3 cm² Each sector is part of a circle with a radius of 5 cm, and since each angle at the centers is 60° (as the triangle is equilateral), the area of one sector is: Area of one sector =60°/360° × πr2 =1/6 ×π ×52 =25π/6 cm2 The total area of the three sectors is=3×25π/6 =25π/2 cm2 The area between the circles is the area of the equilateral triangle minus the area of the three sectors: Area = 25√3 – 25 π /2 =25(√3 - π /2) Therefore, the area subtended between the three circles is: Area = 25 (√3- π /2) cm²

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