perplexus.info :: Shapes : Concentric Circles and a Chord

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.

See The Solution Submitted by Brian Smith

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Both solutions

| Comment 3 of 7 |

Let O be the common center. Call the point where the perpendicular radius meets the chord X, and XO=x. The distance AO=R and CO=r so that we seek (R^2-r^2)pi.

The order of the points C and D is not specified.

Case 1: Order is A-C-D-B.

CX=6/3=3, AX=7+3=10

x^2+3^2 =r^2

x^2+10^2=R^2

R^2-r^2=100-9=91

Area= 91pi

Case 2: Order is A-D-C-B

CX=6/3=3, AX=7-3=4

x^2+3^2=r^2

x^2+4^2=R^2

R^2-r^2=16-9=7

Area= 7pi

Posted by Jer on 2023-07-19 08:57:10

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