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Question

Question: Three equal circles each of radius r touch one another. The radius of the circle touching all the th...

Three equal circles each of radius r touch one another. The radius of the circle touching all the three given circles internally is –

A

(2 +3\sqrt{3} ) r

B

(2+3)3\frac{(2 + \sqrt{3})}{\sqrt{3}}r

C

(23)3\frac{(2–\sqrt{3})}{\sqrt{3}}r

D

(2 – 3\sqrt{3}) r

Answer

(2+3)3\frac{(2 + \sqrt{3})}{\sqrt{3}}r

Explanation

Solution

Q DDEF is equilateral with side 2r if radius of circum circle DEF is R1, then

Area of DDEF =34\frac{\sqrt{3}}{4} (2r)2 = 3\sqrt{3}r2

3\sqrt{3}r2 = 2r.2r.2r4R1\frac{2r.2r.2r}{4R_{1}} Ž R1 = 2r3\frac{2r}{\sqrt{3}}

\ Radius of the circle touching all the three given circles

= r + R1

= r + 2r3\frac{2r}{\sqrt{3}}= (2+3)r3\frac{(2 + \sqrt{3})r}{\sqrt{3}}