Circles and lines

Geometry Level 2

In this image of 3 circles, two straight lines are tangent to all of them. and they are all tangent to the one laying next to them. The radius of the smallest circle is 2. and the biggest of them has a radius of 6. What is the radius of the middle circle?

2 3 2\sqrt{3} 3 3 4 4 10 3 \dfrac{10}{3} 3 2 3\sqrt{2}

Aleksander Lund by Aleksander Lund , 25, Denmark

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1 solution

Consider the diagram on the left.

Using similar triangles, we have

x − 2 x + 2 = 6 − x 6 + x \dfrac{x-2}{x+2}=\dfrac{6-x}{6+x}

Cross-multiplying and simplifying, we get

( x − 2 ) ( 6 + x ) = ( x + 2 ) ( 6 − x ) (x-2)(6+x)=(x+2)(6-x)

6 x + x 2 − 12 − 2 x = 6 x − x 2 + 12 − 2 x 6x+x^2-12-2x=6x-x^2+12-2x

2 x 2 = 24 2x^2=24

x 2 = 24 x^2=24

x = 24 = x=\sqrt{24}= 2 3 \boxed{\color{plum}2\sqrt{3}}

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