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.

nerdiest Jul 28, 2022

#1**+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.

WhyamIdoingthis Jul 28, 2022

#2**+5 **

I drew one line down from b to d to make a 90 degree angle.

90+50=140

180-140=40^{o}

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

nerdiest Jul 28, 2022

#3**+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 \(\overset{\large\frown}{AB} = {80 \over 360} \times 2 \times 18 \times \pi = \color{brown}\boxed{8 \pi }\)

BuilderBoi Jul 28, 2022

#5**+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 \(\frac{80}{360}(\pi\cdot36)=\boxed{8\pi}\text{~inches}.\)

nerdiest Jul 28, 2022

#7**+4 **

Thank you all for your answers, I was kinda stumped when I reached CPhill's answer, and it was wrong.

TYSM

nerdiest Jul 28, 2022

#8**+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 :)

Melody Jul 29, 2022