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Three circles of radius 1 are externally tangent to each other and internally tangent to a larger circle and externally tangent to a smaller circle. What is the radius of the smaller circle? Express your answer as a common fraction in simplest radical form.

 

 May 29, 2021
 #1
avatar+114089 
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I made some assumptions that look right but would need to be proven.

but the jist of my thinking is this.

 

let x be the radius of the smallest circle

the interval joining the centres of the 2 top circles is 2 units long

the distance from each of those radii to the middle of the tiny circle is  1+x

the angle between those two lines is  120 degrees

 

so now you have an isosceles triangle with sides  1+x, 1+x and 2, with the angle between the congruent sides of 120 degrees.

Now use trigonometry to solve.

 May 30, 2021
edited by Melody  May 30, 2021
 #2
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r = 1 / cos(30º)

 May 31, 2021
 #3
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Oops!cheeky

 

r = 1 / cos(30º) - 1

Guest May 31, 2021
 #4
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Doesn't sound right.

Melody  May 31, 2021

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