Question 1199419
<font color=black size=3>
The tutor @ikleyn has the right idea.
However, she made a slight typo.
Side AB should be 5*sqrt(3) cm long.
Edit: Ignore this paragraph as she has corrected the error.


For any 30-60-90 triangle, we have this template
short leg = x
long leg = x*sqrt(3)
hypotenuse = 2x


In this case x = 5
The short leg of this template is always opposite the 30 degree angle. 
Smallest side opposite the smallest angle.


Since AB = 5*sqrt(3) and BC = 5, it leads to triangle ABC having an area of (25/2)*sqrt(3) square cm. 
I skipped the steps so let me know if you need to see them.


Then solving the equation
(1/2)*r*10 = (25/2)*sqrt(3)
will lead to
r = (5/2)*sqrt(3)


area of the full circle = pi*r^2
area of quarter circle = (1/4)*pi*r^2
area of quarter circle = (1/4)*pi*( (5/2)*sqrt(3) )^2
area of quarter circle = (75/16)pi
area of quarter circle = 4.6875pi
area of quarter circle = 4.6875*3.14
area of quarter circle = 14.71875
The value is approximate because pi = 3.14 is approximate.


Unfortunately none of the answer choices you posted are close to this value. 
It's possible your teacher made a typo, or you might have mixed up problems/answer choices by mistake. 
Please revise carefully.
</font>