akshatgupta87 wrote:In the figure to the right, circle O has center O, diameter AB and a radius of 5. Line CD is parallel to the diameter. What is the perimeter of the shaded region?

a)(5/3)pi+5(sqrt3)
b)(5/3)pi+10(sqrt3)
c)(10/3)pi+10(sqrt3)
d)(10/3)pi+5(sqrt3)
e)(10/3)pi+20(sqrt3)

sqrt=Square Root

Please explain...

Given: CD is || to diameter AB, so x = 30º, which implies angle CBE = 2*30º = 60º
So, angle COE = 120º (angle subtended by arc CE at the center of circle is two times the angle subtended at the circumference.
Hence, length of arc CE = 2(pi)(5)(120)/(360) = 10(pi)/3
Also, angles ACB and AEB will be 90º (angle subtended at the circumference by the diameter is always 90º).
Since, angle ACB = 90º and angle ABC = 30º, so angle BAC = 60º, which means triangle ABC is a 30-60-90 triangle. Hence, ratio of the sides are 1:2:√3
AB:AC:BC = 2aa√3, and a = 5 (as 2a = 10)
BC = 5√3
Perimeter of the shaded region = EB + BC + arc CE = 10√3 + 10(pi)/3

The correct answer is C.

akshatgupta87 wrote:thanks..

You are welcome.