Two Circles and a Square

Geometry Level 3

Two identical touching circles of radius r r share the same tangent. A square of side s s , rests on the tangent and touches the two circles as shown in the diagram above.

If r r is 10 units. Then the side s s of the square is how many units?


The answer is 4.

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Vijay Simha by Vijay Simha , 47, USA

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1 solution

Vijay Simha
May 5, 2015

From the figure shown, form the equation using pythagoras theorem,

2r = 2*sqrt(r^2 - (r-s)^2) + s

Solve the quadratic, for r, which will give you two solutions.

Only one of them is valid which yields s = 0.4*r

3 Helpful 0 Interesting 0 Brilliant 0 Confused

I think I am using the same solution as you....hard to explain through the net haha

Zack Yeung - 6 years, 1 month ago

Easier to do it this way: (10-s/2)²+ (10-s)² = 10² which yields s=4, the only admissible solution.

ajit athle - 1 year, 6 months ago

i still didn't get your solution

Swaraj Shandilya - 6 years, 1 month ago

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The length of the "thick" line on top is 2r

The bottom "thin" line which includes the top side of the square is 2 times sqrt(r^2 - (r-s)^2 ) + s

Equate both and solve the quadratic for r

Vijay Simha - 6 years, 1 month ago

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thanks i got it.

Swaraj Shandilya - 6 years ago

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