Question

an arc of AB of a circle subtends an angle theta radian at centre O of the circle. Given that the area of the sector AOB is equal to one third of the square of the lenght of the arc AB; find value of theta​

Answer 1

*Question:-

What is the value of x when an arc AB of a circle subtend an angle x radian at the centre O of the circle. Given that the area of sector AOB is equal to the square of the length of arc AB.?

*Solution:-

AB is a chord to the circle with center O and radius r. Arc AB subtends an angle x radian at the center. Draw OM ⏊ AB such that AM=BM

⎳AOB=x rad

OA = OB = r

Area of sector AOB = (arc AB)² —-(1)

Arc length of AB = rx

Area of sector AOB = (1/2) r² x

From (1),

(1/2) r² x = (rx)²

x² = x/2

x²-(x/2) = 0

x =0 or x = 1/2

Neglecting trivial solution x=0,

x = 1/2

Ans: Value of x = 0.5 rad