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 Concentric Circles and a Chord (Posted on 2023-07-19)
Two concentric circles are drawn with chord AB drawn on the larger circle.
AB cuts through the smaller circle at points C and D.
It is known that AC=7, CD=6, and DB=7.

What is the area between the two circles?

For comparison, the classic problem What's the area? is a version where the chord tangent to the inner circle.

 Submitted by Brian Smith No Rating Solution: (Hide) For clarity, call the radius of the large circle x and the radius of the small circle y. Then the area we want can be written as pi*(x^2-y^2) Draw the diameter containing point C. Let E be the endpoint closer to C and F be the further endpoint. Then EC=x-y and CF=x+y. Now AB and EF are intersecting chords at point C. Then AC*CB=EC*CF; which simplifies to AC*CB = x^2-y^2. Then the area we want to compute is AC*CB*pi. With the dimensions given there are two possible orders for points A, B, C, and D. As suggested by the problem text one order is A, C, D, B. Then AC=7 and CB=CD+DB=13 which make a final area of 91pi. But as Jer noted, the problem still makes sense when C and D are flipped for an order of A, D, C, B. Then AC=7 and CD=DB-CD=1 which makes a final area of 7pi.

 Subject Author Date Subj Rudyard Nolan 2023-09-04 10:42:59 Puzzle Answer K Sengupta 2023-08-26 23:13:49 Math Joshua Cobbs 2023-08-26 13:35:43 Boost Your Luck: Insider Tips For Success In Online Matka On Matkabull matkabull 2023-07-21 00:39:47 Both solutions Jer 2023-07-19 08:57:10 Solution Larry 2023-07-19 08:22:25 solution Charlie 2023-07-19 08:19:47

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