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# geometry

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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
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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!

r = 1 / cos(30º) - 1

Guest May 31, 2021
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Doesn't sound right.

Melody  May 31, 2021