In the image given below, if the radius of all the large circles is r, then what is the radius of the inner smaller circle?


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Answer:

(√2 - 1)r

Step by Step Explanation:
  1. According to the question, the radius of all the large circles is r.
    Let us assume rn as the radius of the inner smaller circle and connect the radius of all of the circles, as shown below.

    If we look at the figure carefully, we notice that
    AB = BC = 2r
    AC = 2r + 2rn
    Now, in right angle triangle ΔABC
    AC2 = AB2 + BC2
    ⇒ AC2 = (2r)2 + (2r)2
    ⇒ AC2 = 4(r)2 + 4(r)2
    ⇒ AC2 = 8(r)2
    ⇒ AC = r√8
    ⇒ AC = r√(2 × 2 × 2
    ⇒ AC = 2r√2
    ⇒ 2r + 2rn = 2r√2
    ⇒ 2rn = 2r√ 2 - 2r
    ⇒ 2rn = 2r(√2 - 1)
    ⇒ rn =  
    2r(√2 -1)
    2
     
    ⇒ rn = (√2 - 1)r
  2. Thus, we can say that the radius of the inner smaller circle is (√2 - 1)r.

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