Answer:
AC = 6√3 in
Step-by-step explanation:
Finding the length of the chord:
Join OC. Now ΔAOC is an isosceles triangle as OA = OC =radius.
∠A = ∠C = 30.
∠A + ∠C + ∠AOC = 180 {angle sum property of traingle}
30 + 30 + ∠AOC = 180°
∠AOC = 180 -60
∠AOC = Ф = 120°
Find the length of radius using the bellow formula.
[tex]\sf \boxed{\bf Arc \ length = \dfrac{\theta}{180}\pi r}[/tex]
Ф = 120°
Arc length = 4π
[tex]\sf 4\pi =\dfrac{120}{180}*\pi *r\\\\ r =\dfrac{4\pi * 180}{120*\pi }\\\\ r = 6 \ in[/tex]
[tex]\sf \boxed{\bf chord \ length = 2rSin \ \dfrac{\theta}{2}}[/tex]
[tex]\sf b = 2*6*Sin \ \dfrac{120}{2}\\\\ b = 2 *6 * Sin \ 60^\circ\\\\ b = 2 * 6 * \dfrac{\sqrt{3}}{2}\\\\ \b = 6\sqrt{3}[/tex]
[tex]\sf \boxed{\bf AC = 6\sqrt{3} \ in}[/tex]
