Question

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Answer 1

Step-by-step explanation:

We can calculate the vertical distance between the centers of the two circles as follows:

ET + TF = EF

ET/2 + TF/2 = EF/2

ET/2 + TF/2 = 6/2 = 3

Suppose the top circle has a radius of x, and the bottom circle has a radius of y. The sum of the radii and the distance above will equal the vertical distance of the rectangle AB, so we have:

x + 3 + y = 8

x + y = 5

We can then construct the line connecting

the two centers of the circle, which will have a distance of x + y = 5. We can then form a right triangle between the circles’ centers: the hypotenuse connects the centers, and the two legs are the vertical and horizontal components of the distance. The vertical distance was computed to be 3 above, so the horizontal distance will be √(52 – 32) = 4, and thus we have the famous Egyptian 3-4-5 right triangle!

We can then sum the two radii and the horizontal leg distance to get the length of BC.

BC = x + 4 + y

BC = 4 + x + y

BC = 4 + 5

BC = 9

Remarkably the answer is the same regardless of the specific sizes of the two circles! And the key to solving this problem was the ancient Egyptian knowledge of the 3-4-5 right triangle.

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Answer 2

Answer:

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