Consider two concentric circles with radii x and 1 where x < 1. In the annulus, the region between the concentric circles, 6 circles are constructed in the following way. Each of these circles is tangent to the inner circle, the outer circle, and the two adjacent circles. Determine the value of x for which this construction is possible. |
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Solution 1 by Alex Bordyukov, Grosse Pointe South High School
To get the answer I connected the centers of all 6 outer circles that I drew. Then connected them to the center of the inner circle. That way I got 6 equal triangles with all angles equal to 60 degrees. Two of the sides of each equilateral triangle are (x+1)/2 [Note: x+(x-1)/2 = (x+1)/2], and the third side is 1-x. Knowing that the triangle is equilateral, the sides must be equal. So, (x-1)/2 = 1-x, which leads to x = 1/3.
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Solution 2 by Dana Harrison, Grand Rapids Forest Hills Northern High School
You can take 7 pennies and arrange them in this fashion. One penny is in the center and the six other pennies surround the central penny. The pennies are all tangent to each other at their points of contact. Then you would have a large circle that surrounds the outer six pennies. Then the diameter of the large circle is three diameters of the smaller circles interior to it. And all 7 circles are congruent. So, the diameter of the central circle is 1/3 the diameter of the large circle. So, the radius of the inner circle is 1/3 the radius of the outer circle. x=1/3.
Also solved by: Jeeun Cho, Jeff Guo (Grosse Pointe
South H.S.); Andrew Gostine (Grand Rapids Catholic Central
H.S.); David Zhang (Ann Arbor Huron H.S.); Travis Rhynard,
Laura Schmidt, Keith Christle (Shepherd H.S.); Wen Fei Liu,
Samantha Falasa (G. R. Forest Hills Northern H.S.).
Correct solutions but insufficient work: Kim Curtiss, F.J.
Dunevant, Bethany Alger, Noel
Dominick, Beke Hollenbach, Jeff Beltinck, Laura Merritt, Brett Krabill
(Shepherd H.S.).