Three circles of radius r touch each other ex | vedclass

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Question

(B) Let the centers of the three circles be $A, B$,and $C$. Since each circle has radius $r$,the distance between any two centers is $2r$. Thus,$ABC$

Vedclass · Accepted Answer

$\frac{2 + \sqrt{3}}{\sqrt{3}}r$

Vedclass · Answer

(B) Let the centers of the three circles be $A, B$,and $C$. Since each circle has radius $r$,the distance between any two centers is $2r$. Thus,$ABC$ forms an equilateral triangle with side length $2r$.Let $O$ be the centroid of this triangle. The distance from the centroid $O$ to any vertex (e.g.,$A$) is given by $OA = \frac{	ext{side}}{\sqrt{3}} = \frac{2r}{\sqrt{3}}$.Let $R$ be the radius of the large circle that touches all three circles internally. The center of this large circle must also be $O$.The radius $R$ is the sum of the distance from the center $O$ to the center of one of the small circles $(A)$ and the radius of that small circle $(r)$.Therefore,$R = OA + r = \frac{2r}{\sqrt{3}} + r = r \left( \frac{2}{\sqrt{3}} + 1 ight) = \left( \frac{2 + \sqrt{3}}{\sqrt{3}} ight)r$.

Three circles of radius $r$ touch each other externally. What is the radius of the circle that touches all three circles internally?

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