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avatar+1132 

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
avatar+267 
+2

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
avatar+1132 
+5

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
avatar+2453 
+1

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
avatar+1132 
+3

Welp, I guess both of you agree!

nerdiest  Jul 28, 2022
 #5
avatar+1132 
+4

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
 #6
avatar+252 
0

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
 #7
avatar+1132 
+4

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
avatar+117852 
+1

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
avatar+1132 
+4

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
 #10
avatar+117852 
+2

Thanks for letting everyone know nerdiest :)

Melody  Jul 30, 2022
 #11
avatar+1132 
+5

yup!

nerdiest  Jul 30, 2022

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