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
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
Hello, stones44!
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.
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
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.
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