circlesgeometrytrigonometry
The radius of a circle is 21.4 meter. Find the length of the chord subtended by a central angle of 110 degrees 40 minutes and the distance between two parallel chords on the same side of the center subtended by central angles 118 degrees 40 minutes and 52 degrees 20 minutes.
Progress: I know the radius is 21.4 meter, I set up a triangle but my final answer was wrong.
If you assume the center of the circle to be the origin of a coordinate system, then the coordinates of one of your points (point $B$ in the illustration I added to your question) has coordinates $\left(d, \sqrt{r^2-d^2}\right)$. From that you can deduce the angle $\varphi$ between the horizontal $x$ axis and the line connecting that point to the origin. That angle will satisfy
$$\tan\varphi = \frac{\sqrt{r^2-d^2}}{d}$$
I've marked $\varphi$ in the above illustration. Now you can use that to compute the angle, and from the angle the arc length:
\begin{align*} \varphi &= \arctan\frac{\sqrt{r^2-d^2}}{d} \\ a_{AB} &= 2r\varphi = 2r\arctan\frac{\sqrt{r^2-d^2}}{d} \end{align*}
If you compute this using a pocket calculator, make sure you have its angle measurement mode set to radians, since a result in degrees won't work for the conversion from angle to arc length.
arc = CD asin(AB/CD)
(where the angle returned by asin is in radians)
EDIT:
I thought the derivation was obvious, but maybe I should explain it.
Mark the midpoint of AB, call it E.
sin(AOE) = (AB/2)/r where r=CD/2
angle = 2 asin(AB/CD)
arc = r angle = CD asin(AB/CD)
Best Answer
Here is a hint:
You have $\dfrac{a}{r} = \sin \alpha$ and $\dfrac{b}{r} = \cos \alpha$.
You may have to take into account that some of these are half what you are interested in.