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new geometry question

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Arc $$AC$$ is a quarter-circle with center $$B$$. The shaded region $$ABC$$ is "rolled" along a straight board $$PQ$$ until it reaches its original orientation for the first time with point $$B$$ landing at point $$B^{\prime}$$. If $$BC = \frac{2}{\pi}$$ cm, what is the length of the path that point $$B$$ travels? Express your answer in simplest form.

Apr 6, 2021

#1
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First it travels one more radius, then the circumference. then one more radius. So, we have $\frac{2}{\pi}+\frac{\pi*2*\frac{2}{\pi}}{4}+\frac{2}{\pi}=\frac{4}{\pi}+\frac{2*2}{4}=\boxed{1+\frac{4}{\pi}}$

Apr 6, 2021
#2
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Sorry, that is wrong because the point B is supposed to travel on a curved path, not just including the start and endpoints.

Apr 6, 2021
#3
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the radius is 2/pi

so the circumference is  2pi*2/pi = 4

So a quarter of the circumference is 1

2/pi  + 1  +  2/pi  = 4/pi  +1

I get the same answer as Riemann

Apr 7, 2021