My Math Forum Ratio of two central angle to the arc lengths of sides
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December 31st, 2012, 07:36 AM   #1
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Ratio of two central angle to the arc lengths of sides

My math books states the follow. Inside the circle, two central angle are formed. One angle is theta with r radius and s arc length. And another angle is Theta1 with s1 arc length and it says that the ratio of theta to theta1 is equal to s to s1...can you prove me how is that equal? See the picture below to visualize.
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 December 31st, 2012, 07:49 AM #2 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 Re: Ratio of two central angle to the arc lengths of sides From the definition of arc length. $arc \ length \= \ (radius) \cdot (angle \ in \ radians)$ using the definition and your diagram $s \= \ r \theta$ $s_1 \= r \theta _1$ Now divide.
 December 31st, 2012, 08:07 AM #3 Member   Joined: Oct 2012 Posts: 56 Thanks: 0 Re: Ratio of two central angle to the arc lengths of sides I think I know part of that. I'm not sure about the rest. $\frac{\frac{r}{r}}{1+\sqrt{1+\frac{r}{r}^2}}=\sqrt {2-1}=22.5^\cir$
 December 31st, 2012, 08:47 AM #4 Newbie   Joined: Dec 2012 Posts: 3 Thanks: 0 Re: Ratio of two central angle to the arc lengths of sides Well, I am not asking you to find the measurement. I am asking why the rations of thetas to their respective side lengths are equals as stated by my pre-calc book? sorry for confusion and thank agentredlum for the answer!
 December 31st, 2012, 08:56 AM #5 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 Re: Ratio of two central angle to the arc lengths of sides You're welcome. Happy new year and welcome to the forum!
 December 31st, 2012, 09:19 AM #6 Newbie   Joined: Dec 2012 Posts: 3 Thanks: 0 Re: Ratio of two central angle to the arc lengths of sides Thank you and happy new year to you too

 Tags angle, arc, central, lengths, ratio, sides

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