find radius of semicircle in which there are 3 consecutive chords measuring 3, 2 and 1 (forming a quadrilateral with the diameter

[manuel augusto]

I am solving it by the Law of Cosines:

 

[Deleted member 4993]

View attachment 14463 I am solving it by the Law of Cosines:

View attachment 14464

Your derived function is:

f(w) = cos^-1(1 - 9/w) + cos^-1(1 - 4/w) + cos^-1(1 - 2/w) - \(\displaystyle \pi\)

I do not see any elementary algebraic method to find root of this function. You can estimate the root by numerical iteration (Newton's method).

 

[pka]

Your derived function is: I do not see any elementary algebraic method to find root of this function. You can estimate the root by numerical iteration

I too see no elementary solution. However If one looks at this page on circular segments scroll down to just above the section on area we see the formula for the central angle: \(\displaystyle \Theta=2\arcsin\left(\dfrac{{\bf{c}}}{2R}\right)\). Where \(\displaystyle \bf{c}=1,~2,~3\) are the lengths of the corresponding chords. Look at this solution. At least while not elementary is is simpler to follow.

 

[manuel augusto]

Thank you very much to all - I'll continue with the problem following the interesting cues you have given me!