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# Feisty Geometry

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In the circle below, $$\overline{AB} \| \overline{CD}$$$$\overline{AD}$$ is a diameter of the circle, and $$AD = 36^{\prime \prime}$$. What is the number of inches in the length of $$\widehat{AB}$$? Express your answer in terms of $$\pi$$.

CPhill had a try at this, but then again, he's human, and the answer he posted was incorrect.

Jul 28, 2022

#1
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The circumference of the circle is 36pi.

Arc BD is 100 degrees as is arc AC. Arc CD and AB are therefore both 80 degree arcs.

80/360 x 36 pi = $$8\pi$$.

This assumes that the latex symbol you used (\widehat, something I've never seen used outside of noting angles) means what I assume it to mean: arc.

Jul 28, 2022
#2
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I drew one line down from b to d to make a 90 degree angle.

90+50=140

180-140=40o

and that was CPhill's way, but it was wrong...

Jul 28, 2022
#3
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Here's my attempt:

Because $$AB \parallel CD$$$$\angle DAB = \angle ADC = 50^\circ$$.

Now, draw line segment $$CB$$, and label the intersection point $$E$$.

Note that $$\triangle ECD$$ is isosceles, so $$\angle ECD = 50^ \circ$$. This means that $$\angle ECD = \angle AEB = 80^ \circ$$

So, the length of $$\overset{\large\frown}{AB} = {80 \over 360} \times 2 \times 18 \times \pi = \color{brown}\boxed{8 \pi }$$

Jul 28, 2022
#4
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Welp, I guess both of you agree!

nerdiest  Jul 28, 2022
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here

By symmetry, $$\widehat{BD}=\widehat{CA}=100^\circ$$. Furthermore,$$\widehat{AB}=\widehat{CD}$$ , so$$[360^\circ=\widehat{AB}+\widehat{BD}+\widehat{DC}+\widehat{CA}=2\widehat{AB}+200^\circ$$Therefore the arc $$\widehat{AB}$$ measures $$80^\circ$$. Since the diameter of the circle is $$36''$$, the length of the arc is $$\frac{80}{360}(\pi\cdot36)=\boxed{8\pi}\text{~inches}.$$

Jul 28, 2022
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Hmm  the thing is, it may be talking about the major arc instead of the minor arc.

Edit: Nevermind, I thought nerdiest was saying that 8pi was wrong. Sorry! :(

Jul 28, 2022
edited by Voldemort  Jul 28, 2022
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Thank you all for your answers, I was kinda stumped when I reached CPhill's answer, and it was wrong.

TYSM

Jul 28, 2022
#8
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Do you understand now nerdiest?

You are a good student here and you never should be worried about saying clearly when you do, or do not fully understand.

I put an emphasis on the work 'clearly'

Thanks to the three of you who have helped here :)

Jul 29, 2022
#9
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yup! I tried my way first, before I asked about this problem, and discovered it was wrong. But, I do fully understand, now that everyone explained it so well.

nerdiest  Jul 29, 2022
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Thanks for letting everyone know nerdiest :)

Melody  Jul 30, 2022
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yup!

nerdiest  Jul 30, 2022