Sum of the radii

Geometry Level 5

Find the sum of the radii of the 2 blue circles. (correct to 1 decimal place)


The answer is 22.5.

Guiseppi Butel by Guiseppi Butel , 90, Canada

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2 solutions

Guiseppi Butel
Sep 26, 2014

Kn = 1/ Rn

K4 = K1 + K2 + K3 +/- 2Sqrt (K1 K2 + K1 K3 + K2*K3)

R4= 1.29978 and 21.21467

Sum = 22.5

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@guiseppi butel Can you elaborate?

Krishna Sharma - 6 years, 8 months ago

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Research "Descartes' theorem". Very interesting.

Guiseppi Butel - 6 years, 8 months ago

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Yes, Descartes' theorem is interesting, and I wish I knew about it before spending so much time trying to solve this problem from scratch. :)

Brian Charlesworth - 6 years, 8 months ago

Descartes theorm is quite too much interesting!

Abhishek Singh - 6 years, 7 months ago
Steven Lee
Jan 4, 2015

I didn't know about Descartes' Theorem so I just used Heron's Formula... You can create a triangle comprised of 3 other triangles to solve for the smaller circle and hence use Heron's formula to get two equations of a single variable (radius) that are equal to each other. As for the bigger circle, you can find a four sided with shape with 3 numerically known lengths and 3 lengths relative to r to again use Heron's Formula to find another relation.

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