Circle Geometry - Tough Question

[kuiperbelt]

I have the answer but how on earth was it arrived at??

https://ibb.co/qyBzJ4w

[Fibonacci]

Question for you here:
https://www.theanswerbank.co.uk/Quizzes-and-Puzzles/Question1838946.html

[PeterT84]

Missed answering question on your thread from Monty Miniman

[Fibonacci]

Thats on the thread Fibonacci mentioned.

[ZebuSanctuary]

Diameter of the smaller circle = 65cm.

[ClareTG0ld]

I agree with Zebu. As to how, draw a couple of right-angled triangles (diagram to be prepared later possibly, but to construct this yourself, draw a straight line from x to g, and a second one from g to a), and use these to generate the equations (capital letters = property of larger circle, small letters similarly):

(1) d = R + (R - 16)
(2) d^2 = 2(R - 9)^2 + R^2 + (R - 16)^2

With a bit of algebra, this reduces to R(R-16) = (R-9)^2, or 2R = 81, from which d = 65 follows.

It may or may not be interesting to have the general results

d = (2a^2 - 2ab + b^2) /(2a - b) = 2r
R = a^2/(2a - b)

where a is the shorter of the two given measurements in the diagram (ie, a is the length of the line EF and b the length of the line AB).

It may also be useful to know that if the outer circle has equation x^2 + y^2 = R^2, then the inner circle has equation

(x + (R - r))^2 + y^2 = r^2

for the definitions of R and r above.

https://www.desmos.com/calculator/farp3mu2ht

[ZebuSanctuary]

A different approach using the geometric mean (gm) associated with 3 numbers. Example given below;

It can be seen that 12 is the gm of 6 and 24 ---> √6x24 = √144 = 12

Alternatively 12² = 6x24 ----> Eq 1

From the circle we can deduce the following;

oa = r - 16
oe = r - 9
ob = r

For some radius r, r - 9 will be the gm of r and r - 16

From Eq 1 we formulate the expression (r - 9)² = r(r - 16)

r² - 18r + 81 = r² - 16r ---> Rearranging;

2r = 81 where 2r is the diameter of the larger circle.

Therefore 2r - 16 = the diameter of the smaller circle i.e., 65 cm

[ClareTG0ld]

I don't entirely understand why your question asked how the answer was arrived at if your BA goes to an answer that does not explain that -- especially when you already had the answer apparently. Zebu's second answer, invoking the geometric mean, seems far more worth attention, although perhaps could do with more justification as to why the GM is relevant here (and I confess I had hoped that providing a diagram including fully manipulable demonstration of all related problems would get some attention).