Answer:
Option D is correct.i.e., ∠ RST = 40°
Step-by-step explanation:
Given: Radius of circle, r = 4 cm
Center of circle is S and Length of Major ARC RT = [tex]\frac{64}{9}\times\pi[/tex]
To find: ∠ RST
Figure is attached
Let [tex]\theta[/tex] be the ∠ RST.
we use the formula of Length of Arc,
[tex]Length\: of\: Arc\: =\:\frac{\theta}{360^{\circ}}\times2\pi r[/tex]
We use this formula to find the angle made by arc RT at centre S,
[tex]Length\:of\:major\:Arc\:=\:\frac{360^{\circ}-\theta}{360^{\circ}}\times2\pi r[/tex]
[tex]\frac{64}{9}\times\pi=\:\frac{360^{\circ}-\theta}{360^{\circ}}\times2\pi \times4[/tex]
[tex]\frac{64}{9}\times\pi=\:\frac{360^{\circ}-\theta}{360^{\circ}}\times8\pi[/tex]
[tex]\frac{360^{\circ}-\theta}{360^{\circ}}=\frac{64\times\pi}{9\times8\pi}[/tex]
[tex]\frac{360^{\circ}-\theta}{360^{\circ}}=\frac{8}{9}[/tex]
[tex]360^{\circ}-\theta=\frac{8\times360}{9}[/tex]
[tex]360^{\circ}-\theta=8\times40[/tex]
[tex]360^{\circ}-\theta=320[/tex]
[tex]-\theta=320-360[/tex]
[tex]-\theta=-40[/tex]
[tex]\theta=40[/tex]
Therefore, Option D is correct.i.e., ∠ RST = 40°
