Find radius and height

Question

I have the following problem: given the length of the chord AB and the length of the arc AB, find the radius of the circle and the height of the triangle ACB where C is a point on the circle such that the height has the maximum length (I mean C is the "most extreme point - the distance from C to AB is maximum as in the figure - the point C is where I wrote $L_{AB}=84$). Any solution? Thanks! Lenth of chord = 74, length of arc=84.

Answer 1

Hint: let $\theta$ be the angle $AOB$. Then $R\theta = L_{AB}=84$ and $2R\sin(\theta/2)=AB=74$.

Height of triangle $ACB$ is $R-R\cos(\theta/2)$.

Answer 2

Your 'extreme height of a triangle' is called an arc's sagitta.
See http://en.wikipedia.org/wiki/Sagitta_(geometry)