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Thanks to David for the suggestion!

If the large circle has a radius equal to 12, what is a + b + c?

As usual, watch the video for a solution.

Circles In A Circle

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“All will be well if you use your mind for your decisions, and mind only your decisions.” Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon.

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Answer To Circles In A Circle

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

The diameter 2b is a radius of the largest circle, so 2b = 12 and b = 6.

To calculate a, construct a right triangle whose hypotenuse connects the centers of the left and middle circles, with length equal to a + b. The legs are equal to b and 12 – a.

Thus we have:

(a + b)2 = b2 + (12 – a)2

Substituting b = 6, we have:

(a + 6)2 = 62 + (12 – a)2
a2 + 12a + 36 = a2 + 144 – 24a + a2
36a = 144
a = 3

The calculation of c is similar but with one more step. Construct a similar triangle between the centers of circles of radius b and c to form a right triangle with hypotenuse b + c. One leg is bc and let the other leg be equal to x.

We will also construct another right triangle. Connect the centers of the largest circle and the smallest. This has length equal to 12 – c, as it is the difference of radius lengths of the largest and the smallest circles. We can then construct a right triangle with one leg x and the other leg c.

Solve for x2 in both right triangles, and substitute b = 6.

Blue triangle
x2 = (b + c)2 – (bc)2
x2 = 4bc
x2 = 24c

Green triangle
x2 = (12 – c)2c2
x2 = 144 – 24c

Set the two equations equal to each other to get:

24c = 144 – 24c
48c = 144
c = 3

Thus we have:

a + b + c
= 4 + 6 + 3
= 13

Special thanks this month to:

Kyle
Lee Redden
Mike Robertson
Daniel Lewis

Thanks to all supporters on Patreon!

Further reading

Brilliant circles
https://brilliant.org/wiki/circles/

Math StackExchange post
https://math.stackexchange.com/questions/3166359/circle-inscribed-in-a-semicircle

Published by

PRESH TALWALKAR

I run the MindYourDecisions channel on YouTube, which has over 1 million subscribers and 200 million views. I am also the author of The Joy of Game Theory: An Introduction to Strategic Thinking, and several other books which are available on Amazon.

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