Circle with radius of 10 in

Chord AB is 14 in

Find measure of minor arc AB and then actual length of minor arc AB

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- Jun 2nd 2007, 03:23 PMstones44Another Circle
Circle with radius of 10 in

Chord AB is 14 in

Find measure of minor arc AB and then actual length of minor arc AB - Jun 2nd 2007, 03:30 PMJhevon
i'm a bit rusty on my circle theorems, so there may be an easier way to do this--but this is the first idea that came to me. see the diagram below

we can use the law of cosines to find the angle x. once we have that, we can use the length of arc formula to find the desired arc - Jun 2nd 2007, 03:46 PMSoroban
Hello, stones44!

Here's another way to determine that central angle $\displaystyle x = \angle AOB$.

In Jhevon's diagram, draw the apothem to chord AB.

Then: .$\displaystyle \sin\left(\frac{x}{2}\right) \:=\:\frac{7}{10}$

- Jun 2nd 2007, 03:54 PMJhevon
- Jun 2nd 2007, 05:19 PMPlato
While I agree Soroban's is standard and nice, I prefer the way you approached the problem.

Look: $\displaystyle x = \arccos \left( {\frac{{200 - 14^2 }}{{200}}} \right)$.

That is certainly calculator ready; and the actual arclength is 10x.

Don't sell yourself short. - Jun 3rd 2007, 09:59 AMSoroban
Hello, Jhevon!

Quote:

As always Soroban does it better.

I certainly am not "better" or "smarter" than you . . . just*lazier.*

. . I was not born with this gift; it took years of dedicated practice.

- Jun 3rd 2007, 02:10 PMstones44
so actual arc length?

how is it 10x? - Jun 3rd 2007, 02:26 PMSoroban
Hello, stones44!

Quote:

so actual arc length?

how is it 10x?

The arc length formula is: .$\displaystyle s \:=\:r\theta$

. . where $\displaystyle r$ is the radius and $\displaystyle \theta$ is the central angle in*radians*.

Using Jhevon's approach: .$\displaystyle \cos x \:=\:0.02\quad\Rightarrow\quad x \:\approx\:1.55$ radians

Therefore: .$\displaystyle s \:=\:10(1.55) \:=\:15.5$ inches.

- Jun 3rd 2007, 05:34 PMstones44
is there another way because I haven't learned that way

- Jun 3rd 2007, 05:40 PMJhevon
- Jun 4th 2007, 04:16 AMtopsquark
- Jun 4th 2007, 07:41 AMJhevon
10 is the radius. and $\displaystyle x$ is the angle that subtends the arc. it may have been better for me to call it $\displaystyle \theta$. so you see the arc length formula is $\displaystyle s = r \theta$, and that's why it is 10x, since $\displaystyle r = 10$ and $\displaystyle \theta = x$. there is no other method as far as this part is concerned. if it is everything up to this point that you are unclear on, please be specific as to what the problem is

- Jun 4th 2007, 08:11 AMPlato
Do you know the origin of the word

*radian*? How large is one radian?

One radian is the measure of the central angle in any circle that intercepts an arc of the circle equal in length to the radius of the circle. So in any circle, if we have a central angle of 2.5 radians the intercepted arc has length 2.5r. If the circle has radius 10 then any central angle of .35 radians intercepts an arc of length 3.5. - Jun 4th 2007, 02:24 PMstones44
but if the measure of the arc is 88.8, meaning the central angle is 88.8, does that mean the length of the arc is 888? (with radius 10)

- Jun 4th 2007, 02:37 PMJhevon
no. you have the angle given in degrees here. you would use a different formula for that. in a circle, there are $\displaystyle 2 \pi$ radians, and no more. anymore than that and it means you are making more than 1 revolution around the circle.

For degrees:

The length of arc $\displaystyle l$ is given by:

$\displaystyle l = \frac { \theta}{360} 2 \pi r$

where $\displaystyle \theta$ is the angle that subtends the arc in degrees