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 = \(\frac{64}{9}\times\pi\)
To find: ∠ RST
Figure is attached
Let \(\theta\) be the ∠ RST.
we use the formula of Length of Arc,
\(Length\: of\: Arc\: =\:\frac{\theta}{360^{\circ}}\times2\pi r\)
We use this formula to find the angle made by arc RT at centre S,
\(Length\:of\:major\:Arc\:=\:\frac{360^{\circ}-\theta}{360^{\circ}}\times2\pi r\)
\(\frac{64}{9}\times\pi=\:\frac{360^{\circ}-\theta}{360^{\circ}}\times2\pi \times4\)
\(\frac{64}{9}\times\pi=\:\frac{360^{\circ}-\theta}{360^{\circ}}\times8\pi\)
\(\frac{360^{\circ}-\theta}{360^{\circ}}=\frac{64\times\pi}{9\times8\pi}\)
\(\frac{360^{\circ}-\theta}{360^{\circ}}=\frac{8}{9}\)
\(360^{\circ}-\theta=\frac{8\times360}{9}\)
\(360^{\circ}-\theta=8\times40\)
\(360^{\circ}-\theta=320\)
\(-\theta=320-360\)
\(-\theta=-40\)
\(\theta=40\)
Therefore, Option D is correct.i.e., ∠ RST = 40°
