Express the length L of the chord of a circle, with given radius r=10cm as a function of the arc length s.

Question

Answer with solution and explanation. Thank you in advance!

Answer 1

The arc length is s=rθ, where r is the radius=10cm, and θ is the subtended angle. Therefore θ=s/10 radians.

The chord length L forms the base of an isosceles triangle with vertex angle θ. We can split the triangle using the perpendicular bisector from the vertex to the base. We now have two congruent right triangles with hypotenuses length r=10 and vertex angle ½θ=s/20. Using trigonometry we have ½L/10=sin½θ=sin(s/20).

So L=20sin(s/20) cm.