Geometry problem with circles

[sheffy2shellfish]

I also used constructing a smaller equilateral triangle below as suggested in the tips video, the method being one with which I also got r = 0.93, R= 12.93 (2d.p.), (also uses Pythagoras and still was wrong). Ik above method where I used Pythagoras twice seems a bit long and there are probably quicker ways, this was just the way I was most familiar with from attempting it this way first, then again you shouldn't need anything hard to solve this it's pretty early geometry. If you arrive at the same answer then it's likely a dodgy website (it's called integral we use it for school, early A level Maths), but the site says: r = 0.71 R = 12.71 so if you get that please send a method and or correct my mistake as I'd like to know what I'm wrong about. Thanks.

 

[JeffM]

I have not tried to work through your answer to see whether it is incorrect. Let the centers of the three medium circles be A, B, and C, and let the center of the little circle be D.

Draw triangles ADB, BDC, and CDA. Notice that they are congruent and isosceles. Why?

Now consider the sum of angles ADB, BDC, and CDA. What is that sum?

Now you know the measure of the angle of ADB and the length of AB.

What is the measure of angle ABD?

Do you know the law of sines?

 

[blamocur]

I could not understand your solution, but I get the same answer, i.e., $r=\frac{6}{\cos 30^\circ} - 6 \approx 0.9282$.

 

[Dr.Peterson]

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I also used constructing a smaller equilateral triangle below as suggested in the tips video, the method being one with which I also got r = 0.93, R= 12.93 (2d.p.), (also uses Pythagoras and still was wrong). Ik above method where I used Pythagoras twice seems a bit long and there are probably quicker ways, this was just the way I was most familiar with from attempting it this way first, then again you shouldn't need anything hard to solve this it's pretty early geometry. If you arrive at the same answer then it's likely a dodgy website (it's called integral we use it for school, early A level Maths), but the site says: r = 0.71 R = 12.71 so if you get that please send a method and or correct my mistake as I'd like to know what I'm wrong about. Thanks.

There definitely are quicker ways, but good work on getting the right answer!

What I did started much as you did, with

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but then I used the fact that this is similar to the 30-60-90 triangle you showed earlier,

​

so that (6+r)/6 = 12/(6 sqrt(3)).

From that you can get r immediately.

You just skidded past the right turn, and went around the block an extra time! (Looking back and seeing what you could have done is how we all learn to see quicker methods the next time!)