Answer
$$L=20 \sin \frac{ \theta }{2}$$
Work Step by Step
To find the length $L$, one should first draw a line from the center of the circle to the chord, perpendicular to it. The resultant right triangles are congruent to each other since one of their sides is one radius of the circle and one side is common and so their other sides become equal to each other (by applying the Pythagorean Theorem). Since the triangles congruent to each other, we can conclude that the common side bisects both the chord and the angle $\theta$. So we have$$\frac{L}{2}= 10 \sin \frac{ \theta }{2} \quad \Rightarrow \quad L=20 \sin \frac{ \theta }{2}.$$We have used the fact that in any right triangle $\sin \theta$ equals the ratio of length of the opposite side to the ratio of length of the hypotenuse.
